Soft Noncommutative Schemes Via Toric Geometry and Morphisms From

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June 2021 arXiv:yymm.nnnnn [math.AG] D(15.1), NCS(1): D-brane on soft NC space

Soft noncommutative schemes via toric geometry and morphisms from an Azumaya scheme with a fundamental module thereto — (Dynamical, complex algebraic) D-branes on a soft noncommutative space

Chien-Hao Liu and Shing-Tung Yau

Abstract

A class of noncommutative spaces, named soft noncommutative schemes via toric geometry, are constructed and the mathematical model for (dynamical/nonsolitonic, complex algebraic) D-branes on such a noncommutative space, following arXiv:0709.1515 [math.AG] (D(1)), is given. Any algebraic Calabi-Yau space that arises from a complete intersection in a smooth toric variety can embed as a commutative closed subscheme of some soft noncommutative scheme. Along the study, the notion of soft noncommutative toric schemes associated to a (simplicial, maximal cone of index 1) fan, invertible sheaves on such a noncommutative space, and twisted sections of an invertible sheaf are developed and Azumaya schemes with a fundamental module as the world-volumes of D-branes are reviewed. Two guiding questions, Question 3.12 (soft noncommutative Calabi-Yau spaces and their mirror) and Question 4.2.14 (generalized matrix models), are presented. arXiv:2108.05328v1 [math.AG] 11 Aug 2021

Key words: monoid algebra; soft noncommutative toric scheme, soft noncommutative scheme, invertible sheaf, twisted section; D-brane, Azumaya scheme, morphism

MSC number 2020: 14A22; 14A15, 14M25; 81T30

Acknowledgements. We thank Andrew Strominger, Cumrun Vafa for influence to our understanding of strings, branes, and gravity. C.-H.L. thanks in addition Chin-Lung Wang for communications on Noncommutative Algebraic Geometry during the brewing years; Enno Keßler, Carlos Shahbazi, Peter West for communications related to a concurrent in-progress work during the COVID-19 lockdown year that will cross the current work in the future; Xi Yin and Sebastien Picard for topic courses, spring 2020; Dror Bar-Natan, Galia Dafni, Ayelet Lindenstrauss for a surprise contact; Ren-De Hwa for bringing MuseScore to his attention, spring 2020, just before the lockdown; Pei-Jung Chen for the biweekly communications on J.S. Bach’s sonata and Ling-Miao Chou for the daily exchange of progress and comments that improve the illustrations and the tremendous moral support, which together boost the morale during the lockdown; and the numerous people behind the scene who make the continuation of projects possible during the COVID-19 pandemic times (cf. dedication & continuing acknowledgements). The project is supported by NSF grants DMS-9803347 and DMS-0074329. Chien-Hao Liu dedicates this first work in the second spin-off NCS of the D-project to a group of familiar strangers, including the building common-area cleaners and waste collectors, post-and-parcel delivery-service persons, and workers and staff in the nearby Star Market/Shaw’s, Walgreens, Reliable Market, Inman Square Hardware, CVS Pharmacy, Skenderian Apothecary, post office at Harvard Square, Kinko’s/FedEx at Harvard Square, Bob Slate Stationer, Bank of America at Harvard Square, Cambridge Savings Bank at Harvard Square and Inman Square, Sullivan Tires and Auto Service, and Cambridge Honda, and Christine Rishebarger, Joseph Yager, and their team members, and many more people behind the scene whom he would likely never have a chance to meet for their indispensable work and services in the COVID-19 pandemic times that make his life going on as normally as it can be; and to Lynda Azar, who served in the Condominium Board for over a decade to promote a lot of improvement but sadly had to change her course of life due to the COVID-19 pandemic.

Continuing acknowledges from C.-H.L.: Since ancient times death has been a taboo in many cultures. Yet, no matter how glorious one may be through one’s life, at the end one still has to face one’s ﬁnal fate and last journey: death, which one can only face alone. Having lived a very simple life in the United States for decades, naturally I would not expect this to be a pressing issue that I have to think about right now, despite that every incident of death of people I had been close to and cherished in my life always provoked some deep reﬂections. Then came the COVID-19 pandemic outbreak, spring 2020, in the United States. When one read the daily statistics of deaths and lived in a situation that essential items in grocery stores had to be rationed, one was immediately submerged into an atmosphere of a seemingly wartime and the issue of death looked indeed very near and real. What work or deed or achievement or honor would still retain its meaning in front of death? This is a counter-question to the question of the meaning of life, which we mostly don’t bother to think about in a peaceful time. I would like to thank in addition the following authors, lecturers, musicians, developers, creators, artists, and hosts · Kelli Francis-Staite, Dominic Joyce; Daniel S. Freed; Mark Gross; Arthur Ogus; Costas Bachas; Sergio Ferrara; Amihay Hanany; Kentaro Hori, Amer Iqbal, Cumrun Vafa; Gregory W. Moore; Joseph Polchinski†; Leonard Susskind; Steven Weinberg† · William Christ, Richard DeLone†, Vernon Kliewer†, Lewis Rowell, William Thomson; Craig Wright · Arthur H. Benade†; Michel Debost, Jeanne Debost-Roth; Seta Der Hohannesian; John C. Krell†; Hui-Hsuan Liang; Jia-Hua Lin; Yi-Hui Lin; David McGill; Emmanuel Pahud; Amy Porter; Erwin Stein†; William Vennard†; Xiao-Jen Wu · [MuseScore] Thomas Bonte, Nicolas Froment, Werner Schweer, plus volunteers; [Audacity] Roger Dannenberg, Dominic Mazzoni, plus volunteers · Michael Schuenke, Eric Schulte, Udo Schumacher: Markus Voll, Karl Wesker: Anne M. Gilroy, Brian R. MacPherson, Lawrence M. Ross · Arthur C. Guyton†, John E. Hall, Michael E. Hall · Charles Kuralt†; Juan Li; Ying Liu; Quan-Chong Luo†; Qingchun Shu (Lao She)†; Ann Voskamp; Rui-Hong Xia; Malala Yousafzai · Robert Barrett; George Bridgman†; Andrew Loomis†; Frank Reilly†; John Singer Sargent†; Mau-Kun Yim, Iris Yim // Danqing Chen; Susie Hodge // Sarah Simblet; Uldis Zarins · Yu-Han Ho; Sarah Ivanhoe; Jennifer Kries; Yu-Han Liu; Yo-Tse Tsai · Cheng Hsuan; Brij Kothari; Wei-San Wang · Waldemar Januszczak; Ziqi Li; program hosts of 99.5 WCRB - Classical Radio Boston for their book or book series, works, lecture series, master classes, demonstrations, freewares, creations, life stories, programs, or intellectual legacies that provided intellectual and physical challenges, inner growth, a tool to access some of the most creative minds crossing over eight hundred years from Bernart de Ventadorn (1135-1194) to Aaron Copland (1900-1990), and the joy of learning and reading for a life in social distancing and isolation, immersed me in the various categories of arts and beauties humankind had ever created, and made this accidental soul-searching journey that accompanied the brewing and preparation of the current and other in-progress work less scary while under the shadow of death. (†: deceased; with salute from CHL) Soft Noncommutative Schemes and Morphisms from Azumaya Schemes Thereto

0. Introduction and outline The realization of the noncommutative feature of D-brane world-volumes ([H-W], [Po1], [Wi]; [L-Y1], [Liu]) makes D-branes a good candidate as a probe to noncommutative geometry. Un- fortunately, all the three building blocks of modern Commutative Algebraic Geometry — the notion of localizations of a ring, the method of associating a topology to a ring via Spec, and the approach to understand a ring by studying its category of modules — have their limitation when extending to noncommutative rings and Noncommutative Algebraic Geometry. (See, e.g., [Ro1], [Ro2], [B-R-S-S-W] to get a feel.) Unlike a (fundamental) superstring world-sheet, which can have a very abundant class of geometry as its target-spaces —most notably Calabi-Yau spaces — it is not immediate clear what class of noncommutative spaces can serve as the target-space of a dynamical super D-string world-sheet and how to construct them, let alone the even deeper issue of Mirror Symmetry phenomenon when two different target-spaces can give rise to isomorphic 2-dimensional supersymmetric quantum field theories on the superstring world-sheet. In this work we introduce a class of noncommutative spaces, named ‘soft noncommutative schemes’ via toric geometry (Sec. 2 and Sec. 3), to partially take care of the persistent difficulty of gluing in general Noncommutative Algebraic Geometry. This by construction is only a very limited class of noncommutative spaces, yet abundant enough that all the algebraic Calabi- Yau spaces that arise from complete intersections in a smooth toric variety can show up as a commutative subscheme of some soft noncommutative scheme (Corollary 3.11). A mathematical model for (dynamical, complex algebraic) D-brane on such a noncommutative space is then given (Sec. 4) as morphisms from Azumaya schemes with a fundamental module thereto. From this, one is certainly very curious as to how Mirror Symmetry stands when these soft noncommutative spaces are taken to be the target-spaces of D-string world-sheets. Convention. References for standard notations, terminology, operations and facts are (1) Azumaya/matrix algebra: [Ar], [Az], [A-N-T]; (2) commutative monoid: [Og]; (3) toric geometry: [Fu]; (4) aspects of noncommutative algebraic geometry: [B-R-S-S-W]; (5) commutative algebra and (commutative) algebraic geometry: [Ei], [E-H], [Ha]; (6) string theory and D-branes: [Ba], [Jo], [Po1], [Po2], [Sz]. · All commutative schemes are over C and Noetherian. · For a sheaf F on a gluing system of charts, the notation ‘s ∈ F’ means a local section s ∈ F(U) for some chart U.

Outline

0 Introduction 1 1 Monoids and monoid algebras 2 2 Soft noncommutative toric schemes 4 2.1 The noncommutative aﬃne n-space ncAn and its 0-dimensional subschemes ...... 5 C 2.2 Soft noncommutative toric schemes associated to a fan ...... 8 3 Invertible sheaves on a soft noncommutative toric scheme, twisted sections, and soft noncommutative schemes via toric geometry 15 4 (Dynamical) D-branes on a soft noncommutative space 21 4.1 Review of Azumaya/matrix schemes with a fundamental module over C ...... 22 4.2 Morphisms from an Azumaya scheme with a fundamental module to a soft noncommutative scheme Y˘ : (Dynamical) D-branes on Y˘ ...... 23 4.3 Two equivalent descriptions of a morphism ϕ : XAz → Y˘ ...... 30

1 1 Monoids and monoid algebras

Basic definitions of monoids and monoid algebras we need for the current work are collected in this section to fix terminology and notation. Readers are referred to [De], [Ge], [G-H-V] and [Og], from where these definitions are adapted, for more details.

Definition 1.1. [monoid] A monoid is a triple (M, ?, eM ), abbreviated by (M,?) or M, that consists of a set M, an associative binary operation ?, and a (two-sided) identity element eM of (M,?): eM ? m = m ? eM = m for all m ∈ M. An element m ∈ M is invertible if there 0 0 0 0 exists an element m ∈ M such that m ? m = m ? m = eM . When m is invertible, such m is unique and is denoted m−1, called the inverse of m. The center Center (M) of M is the submonoid {c ∈ M | c ? m = m ? c, for all m ∈ M} of M.A homomorphism of monoids (or monoid homomorphism) is a function θ : M → N between monoids such that θ(eM ) = eN and θ(m ? m0) = θ(m) ? θ(m0). θ is called a monomorphism if it is injective, and an epimorphism if surjective. Given a set S, the free monoid F (S) associated to S is the monoid that consists of an element e and all formal words s1 ··· sk of finitely many elements of S, with the product rules 0 0 0 0 s1 ··· sk ? s1 ··· sl = s1 ··· sks1 ··· sl and e ? s1 ··· sk = s1 ··· sk ? e = s1 ··· sk. Let S ⊂ M be a subset of a monoid. We say that M is generated by S if the monoid homomorphism F (S) → M specified by the inclusion S,→ M, with e 7→ eM and s1 ··· sk 7→ s1 ? ··· ? sk, is surjective. We say that a monoid M is finitely generated if M is generated by a finite subset S ⊂ M. We say that a generating set S of M is inverse-closed if s ∈ S and s is invertible in M, then s−1 ∈ S. By convention, when a monoid is written multiplicatively and the monoid operation ? is clear 0 0 0 from the content, m1 ? m will be denoted simply m · m or mm . Commutative monoids are often written additively, with ? denoted + and eM denoted 0.

Deﬁnition 1.2. [product and central extension] Let (C, ·, 1) and (M, ?, eM ) be monoids. The product of (C, ·, 1) and (M, ?, eM ) is the monoid with elements (c, m) ∈ C × M with (c1, m1) ∗ (c2, m2) = (c1 · c2, m1 ? m2) and identity (1, eM ). When C is commutative, we’ll denote the product C · M or CM and elements cm . In this case, C,→ C · M →→ M, with c 7→ c eM and cm 7→ m, is a central extension of M.

Definition 1.3. [Cayley graph of monoid with finite generating set] Let (M, ?, eM ) be a monoid with an inverse-closed, finite generating set S. Then the Cayley graph ΓCayley (M,S) of (M,S) is a directed graph with the set of vertices M and, for every pair (m1, m2) of vertices a directed edge labelled by s ∈ S from m1 to m2 if m2 = m1 ? s. We take the convention that if both s and s−1 are in S, then the edge associated to s and that associated to s−1 coincide with opposite directions; (i.e. the corresponding edge is bi-directed). Since every m ∈ M can be expressed as a finite word in letters from S,ΓCayley (M,S) is a connected graph.

Definition 1.4. [edge-path monoid] (Continuing Definition 1.3.) Endow the Cayley graph Γ := ΓCayley (M,S) of (M,S) with the topology from the geometric realization of Γ as a 1- dimensional simplicial complex. An edge-path at the vertex m0 on Γ, with the terminal vertex mt, is a direction-preserving continuous map γ : [0, 1] → Γ, with γ(0) = m0 and γ(1) = mt, that is piecewise linear after a subdivision of the interval [0, 1] as a 1-dimensional simplicial complex. Thus, associated to each edge-path γ at m0 is a word wγ = s1 ··· sk in letters from S, where s1 is the initial edge γ takes from the vertex m0 and sk is the final edge γ takes to the terminal

2 vertex mt. Given an edge-path γ at m0 and a vertex m ∈ M, there is a translation Tm : γ 7→ mγ, where mγ is the edge-path at m that takes the same word wγ as γ and is piecewise linear under the same subdivision of [0, 1]. The constant edge-path γ : [0, 1] → Γ such that γ([0, 1]) = m ∈ M will be denoted by the vertex m. There is an operation ? on the set of edge-paths:

· For edge-paths γ1 and γ2 on Γ such that γ1(1) = γ2(0), deﬁne

( 1 γ1(2t) , for t ∈ [0, ]; (γ ? γ )(t) = 2 1 2 1 γ2(2t − 1) , for t ∈ [ 2 , 1] .

· For general edge-paths γ1 and γ2 on Γ, deﬁne

γ1 ? γ2 := γ1 ?Tγ1(1)(γ2) .

In particular, m ? γ = Tm(γ), γ(0) ? γ = γ, and γ ? m = γ, after a subdivision of [0, 1] and a simplicial map on the subdivided [0, 1] that ﬁxes {0, 1}, for all vertex m and edge-path γ. Consider the space M˘ of homotopy classes of edge-paths at the vertex eM of Γ relative to the boundary {0, 1} of the interval [0, 1]. The operation ? on edge-paths on Γ descends to an associative binary operation, still denoted by ?, on M˘ with the identity the constant path eM . The monoid (M,˘ ?, eM ) will be called the edge-path monoid associated to (M,S).

Deﬁnition 1.5. [monoid algebra] Let (M, ?, eM ) be a monoid. The C-vector space L m∈M C · m with the binary operation ∗ generated by C-linear expansion of

(c1m1) ∗ (c2m2) = (c1c2) · (m1 ? m2) is called the monoid algebra over C associated to M, in notation ChMi and the identity 1·eM =: 1. When M is commutative, we will denote ChMi by C[M].

Lemma 1.6. [natural homomorphism] There is a built-in monoid epimorphism π : M˘ → M, which extends to an epimorphism π : ChM˘ i → ChMi of monoid algebras over C.

Proof. This follows from the surjective map π : {edge-paths γ at eM } → M, with γ 7→ γ(1).

n Example 1.7. [lattice of rank n] Let M be the lattice of rank n (i.e. ' Z as a Z-module). n Z As a commutative monoid written multiplicatively, M ' ×i=1zi and generated by the inverse- −1 −1 closed subset S = {z1, z1 , ··· , zn, zn }. The associated C-algebra is a polynomial ring 0 0 −1 −1 C[z1, z1, ··· , zn, zn] C[M] = C[z1, z , ··· , zn, zn ] ' 0 0 . (z1z1 − 1, ··· , znzn − 1) The Cayley diagram Γ of (M,S) is a homogeneous graph of constant valence 2n for all vertices, with the base vertex 1 ∈ M. Under the above presentation of M, the edge-path monoid M˘ is −1 −1 −1 −1 hz1, z1 , ··· , zn, zn i (the noncommutative monoid generated by z1, z1 , ··· , zn, zn subject to −1 −1 relations zizi = zi zi = 1, i = 1, . . . , n and the associated monoid algebra 0 0 ˘ −1 −1 Chz1, z1, ··· , zn, zni ChMi = Chz1, z1 , ··· , zn, zn i ' 0 0 0 0 . (z1z1 − 1, z1z1 − 1, ··· , znzn − 1, znzn − 1)

3 γ2

Γ 1

γ1

π([ γ 2 ])

z2 1 z1 M

π([ γ 1 ])

Figure 1-1. The edge-path monoid M˘ associated to a lattice M (at the bot- tom) with generators indicated by the dashed grid. The rank 2 case is illustrated here. All the edges of the Cayley graph Γ (at the top) of M-with-generators are bi-directed. Examples of two elements of M˘ , represented by edge-paths γ1 = 3 2 −2 −1 2 −2 −1 −1 −1 −3 −2 −1 −1 −1 3 −2 2 −1 z1z2 z1 z2 z1 z2z1 z2 z1 z2 z1z2 z1 and γ2 = z2 z1 z2z1 z2z1 z2 z1z2 z1 z2 at the identity 1 ∈ Vertex(Γ), and their image under the built-in monoid homomorphism ˘ −2 π : M → M are indicated: π([γ1]) = z1z2 and π([γ2]) = 1.

0 0 Here, Chz1, z1, ··· , zn, zni is the free associative C-algebra with 2n generators and 0 0 0 0 (z1z1 − 1, z1z1 − 1, ··· , znzn − 1, znzn − 1) is the two-sided ideal generated by the elements indicated. The monoid homomorphism π : M˘ → M from the terminal-vertex-of-edge-path-at-1 map and the associated monoid-algebra homomorphism π : ChM˘ i → C[M] coincide with the quotient by the commutator ideal [M,˘ M˘ ]. Cf. Figure 1-1. n × n The complex torus T := T := (C ) action on the generators

gt (z1, ··· zn) 7−→ (t1z1, ··· , tnzn) , t = (t1, ··· , tn) ∈ T , n induces a T -action on C · M and C · M˘ , leaving each C-factor of C · M and C · M˘ invariant, n n that render π : C · M˘ → C · M T -equivariant. This extends to a T -action on ChM˘ i and C[M] n under which π : ChM˘ i → C[M] is T -equivariant.

The lattice M of rank n in Example 1.7 is the starting point of constructions of the current work. As a commutative monoid, there are occasions when it is more convenient for M to be written additively, rather than multiplicatively. We shall switch between these two pictures freely and use whichever suits better.

2 Soft noncommutative toric schemes

Soft noncommutative toric schemes are introduced in this section. The setting and the terminology used indicate our focus on the function ring, rather than the point-set and topology, of a noncommutative space while striving to retain enough underlying geometric picture for the purpose of studying (dynamical) D-branes on a noncommutative target-space in string theory within the realm of Algebraic Geometry. (Cf. [L-Y1] (D(1)), [L-L-S-Y] (D(2)), and Sec. 4.2 of the current work.)

4 2.1 The noncommutative affine n-space ncAn and its 0-dimensional subschemes C Definition 2.1.1. [affine scheme, function ring, noncommutative affine n-space over op C] Let Ring be the category of rings. An object X in the opposite category Ring of Ring is called an affine scheme. The ring that underlies X is called the function ring of X. If the function ring of X is noncommutative (resp. commutative), then we say that X is a noncommutative affine scheme (resp. commutative affine scheme). In particular, we define a noncommutative affine n- op space over C to be an object in Ring that corresponds to a free noncommutative associative C-algebra generated by n letters, in notation Chz1, ··· , zni. Since any two free noncommutative associative C-algebras generated by n letters are isomorphic for a fixed n, we shall denote a noncommutative affine n-space over commonly by ncAn or simply ncAn. ncAn is smooth in C C the sense of the following extension/lifting property:

· For any C-algebra homomorphism Chz1, ··· , zni → A and any C-algebra epimorphism B →→ A, there exists a C-algebra homomorphism Chz1, ··· , zni → B that makes the following diagram commute

5 B

Chz1, ··· , zni / A.

Deﬁnition 2.1.2. [master commutative subscheme of ncAn] The natural quotient

Chz1, ··· , zni −→−→ C[z1, ··· , zn] , zi 7−→ zi of C-algebras, with kernel the two-sided ideal (zizj − zjzi | 1 ≤ i < j ≤ n), deﬁnes an embedding An ,→ ncAn, called the master commutative subscheme of ncAn.

n n Deﬁnition 2.1.3. [closed subscheme of ncA ] A closed subscheme of ncA is a C-algebra quotient Chz1, ··· , zni −→−→ A. When A is commutative, then the closed subscheme is called a commutative closed subscheme of ncAn. Since when A is commutative, such a quotient always factors as hz , ··· , z i / / A C 1 n 9 9

( ( C[z1, ··· , zn] , a commutative closed subscheme of ncAn is always contained in the master commutative sub- n n scheme A ⊂ ncA . When A is a ﬁnite-dimensional C-algebra, the closed subscheme is called n n a 0-dimensional subscheme of ncA . In particular, a C-point on ncA is a C-algebra quotient n Chz1, ··· , zni −→−→ C. Since C is commutative, the set of C-points of ncA coincide naturally n n n with the set of C-points of A via the built-in embedding A ,→ ncA .

Deﬁnition 2.1.4. [punctual versus nonpunctual 0-dimensional subscheme] A 0-dimen- n sional subscheme Chz1, ··· , zni →→ A of ncA is called punctual if A admits a C-algebra quotient

A −→−→ Center (A)/Nil Center (A) .

Here, Center (A) ⊂ A is the center of A and Nil Center (A) is the ideal of nilpotent elements of Center (A). Else, the 0-dimensional subscheme is called nonpunctual (i.e. “fuzzy without core”).

5 Example 2.1.5. [Grassmann points, Azumaya/matrix points, other special points on ncAn] (1) All commutative 0-dimensional subschemes of ncAn are punctual. n (2) A Grassmann point (resp. Azumaya or matrix point) on ncA is a C-algebra quotient Chz1, ··· , zni →→ A such that A is isomorphic to a Grassmann algebra over C

anti-c C[θ1, ··· , θr] := Chθ1, ··· , θri/(θαθβ + θβθα | 1 ≤ α, β ≤ r) for some r (resp. a matrix algebra Mr(C) over C for some rank r ≥ 2). Both are noncommutative points on ncAn. However, Grassmann points are punctual while Azumaya/matrix points are nonpunctual since there exists no C-algebra homomorphism from Mr(C) to C for r ≥ 2. n (3) Other special points on ncA include points Chz1, ··· , zni →→ A where A is isomorphic upper to the C-algebra Tr (C) of upper triangular matrix for some rank r ≥ 2 (resp. the C-algebra lower Tr (C) of lower triangular matrix for some rank r ≥ 2.) They are called upper-triangular matrix points (resp. lower-triangular matrix points). Unlike Azumaya/matrix points on ncAn, both are punctual under the correspondence that sends an upper-or-lower triangular matrix to its diagonal.

Deﬁnition 2.1.6. [nested master l-commutative closed subscheme of ncAn] Given two two-sided ideals I,J in Chz1, ··· , zni. Denote by [I,J] the two-sided ideal in Chz1, ··· , zni generated by elements of the form [f, g] := fg − gf, where f ∈ I and g ∈ J. Now let m0 be the two-sided ideal (z1, ··· , zn) of Chz1, ··· , zni. Then note that [m0, m0] and (zizj − zjzi | 1 ≤ i < n n j ≤ n) coincide. Consider the closed scheme A(l) of ncA , l = 1, 2, ··· , associated the C-algebra quotient l Chz1, ··· , zni −→−→ Chz1, ··· , zni/[m0, m0] . 2 3 Since [m0, m0] ⊂ [m0, m0] ⊂ [m0, m0] ⊂ · · · , one has a nested sequence of closed subschemes in ncAn n n n n A = A(1) ,→ A(2) ,→ A(3) ,→ · · · . n Any function on ncA of the form f1 ··· fl, where fi ∈ m0 for all i, becomes commutative when n 0 n restricted to A(l0) for l ≤ l. Furthermore, any closed subscheme of ncA with this property n n must be a closed subscheme of A(l). Thus, we name A(l) the master l-commutative subscheme of n n ncA (with respect to the choice of coordinate functions z1, ··· , zn on ncA ). The restriction n n n of any function f on ncA of degree d ≥ 1 to any A(l) with l ≤ d is commutative on A(l).

n Example 2.1.7.[ 0-dimensional subscheme and A(l)] How a 0-dimensional subscheme in n n ncA intersects the nested sequence A(l), l = 1, 2,..., gives one a sense of a depth of noncommutativity the 0-dimensional subscheme of ncAn is located at. (1) A commutative 0-dimensional n n n subscheme of ncA lies in A = A(1) and hence in A(l) for all l. (2) A punctual 0-dimensional subscheme of ncAn contains a commutative subscheme and n n hence must have nonempty intersection with A and hence has nonempty intersection with A(l) for all l. They thus lie in an inﬁnitesimal neighborhood of An in ncAn. (3) Consider the Azumaya/matrix point on ncAr2

Chzij | 1 ≤ i, j ≤ ri −→−→ Mr(C) , zij 7−→ eij , where eij is the r × r matrix with ij-entry 1 and elsewhere 0. The kernel of this C-algebra epimorphism is the two-sided ideal I generated by

0 0 zijzi0j0 − δji0 zij0 , 1 ≤ i, j, i , j ≤ r , where δ• is the Kronecker delta.

6 In particular, zij − z 0 z 0 0 ··· z 0 0 z 0 ∈ I for all l = 1, 2,.... Recall the two-sided ideal ij1 j1j2 jl−1jl jlj m0 = (zij | 1 ≤ i, j ≤ r) of Chzij | 1 ≤ i, j ≤ ri. Then the two-sided ideal generated by l I + [m0, m ] in Chzij | 1 ≤ i, j ≤ ri is the whole Chzij | 1 ≤ i, j ≤ ri. This shows that this r2 r2 Azumaya point on ncA has no intersection with A(l) for all l. In a sense, this Azumaya point is located deep in the noncommutative part of ncAn.

Deﬁnition 2.1.8. [connectivity and hidden disconnectivity of 0-dimensional sub- n scheme] Given a 0-dimensional subscheme Chz1, ··· , zni →→ A on ncA . We say that the 0-dimensional subscheme is connected if A is not a product A1 × A2 of two C-algebras. Else, we say that the 0-dimensional subscheme is disconnected. If A is not a product, yet it admits a decomposition of 1 by non-zero orthogonal idempotents

1 = e1 + ··· + ek , 2 for some k, where ei 6= 0 and ei = ei for all i and eiej = ejei = 0 for all i 6= j, then we say that the connected 0-dimensional subscheme has hidden disconnectivity.

The phenomenon of hidden disconnectivity can occur for noncommutative points — for example, Azumaya/matrix points, upper-triangular matrix points, lower-triangular matrix points — on ncAn whether they are punctual or not. (Cf. [L-Y1] (D(1)) for hidden disconnectivity of Azumaya schemes via the notion of surrogates.)

Any C-algebra homomorphism ] u : C[t] −→ Chz1, ··· , zni ] with u (t) 6∈ C is a monomorphism and, by deﬁnition/tautology, induces a dominant morphism u : ncAn −→ A1 . n In this way, a 0-dimensional subscheme Chz1, ··· , zni →→ A on ncA is mapped to a 0- dimensional, now commutative, subscheme on the aﬃne line A1. One may use such projections to get a sense as to how the 0-dimensional subscheme sits in ncAn and some of its properties. (Cf. Radom transformation in analysis.)

Example 2.1.9. [hidden disconnectivity of Azumaya/matrix point on ncAn mani- fested via projection to A1] Recall the matrix point on ncAr2 in Example 2.1.7 (3). Consider r2 1 the projection u : ncA → A speciﬁed by the C-algebra monomorphism ] 0 u : C[t] −→ Chzij | 1 ≤ i, j ≤ ri, t 7−→ zi0i0 for some i ; ] and letu ˆ be a C-algebra homomorphism from the composition

uˆ] * C[t] / Chzij | 1 ≤ i, j ≤ ri / Mr(C) u] induced by the given matrix point on ncAr2 . Then the matrix point, though connected, is projected under u to two distinct closed points {p = 0 , p = 1} on A1, described by the ideal ] ] (t(t − 1)), i.e. the kernel Ker (û ) ofu ˆ , of C[t]. In this way, u reveals the hidden disconnectivity r2 ] of the given matrix point on ncA . As a (left) C[t]-module throughu ˆ , Mr(C) is pushed forward 1 2 under u to a torsion OA1 -module on A , with fiber-dimension r − r at p = 0 and r at p = 1. r2 1 For reference, if one instead considers the projection v : ncA → A specified by the C- ] 0 0 algebra monomorphism v : C[t] → Chzij | 1 ≤ i, j ≤ ri with t 7→ zi0j0 for some i 6= j . Then the 1 2 given matrix point is projected to a nonreduced point on A associated to the ideal (t ) of C[t].

7 ncAn as a smooth noncommutative aﬃne toric scheme

n Let N be a lattice isomorphic to Z , with a fixed basis (e1, ··· , en), and M = Hom Z(N, Z) be the dual lattice, with the evaluation pairing denoted by h , i : M × N → Z and the dual basis ∗ ∗ (e1, ··· , en). Denote the associated R-vector spaces NR := N ⊗Z R and MR := M ⊗Z R. ∨ Let σ be the cone in NR generated by e1, ··· en. Then the commutative monoid Mσ := σ ∩M ∗ ∗ ˘ is generated by {e1, ··· , en}. Let M be the associated noncommutative monoid from edge-paths ∗ at 0, cf. Definition 1.4. Then the correspondence ei 7→ zi, i = 1, . . . , n, specifies a C -algebra ˘ n isomorphism from the monoid algebra ChMσi to the function ring Chz1, ··· , zni of ncA . Under the built-in monoid homomorphism π : M˘ σ → Mσ, this isomorphism descends to an isomorphism n C[Mσ] → C[z1, ··· , zn] that gives the standard realization of A as a toric variety. Furthermore, × ˘ × n since the multiplicative group C := C−{0} in C lies in the center of ChMσi, the (C ) -action on × n C[Mσ] generated by (z1, ··· , zn) 7→ (t1z1, ··· , tnzn) for (t1, ··· , tn) ∈ (C ) lifts canonically × n ˘ × n to a (C ) -action on ChMσi that makes π (C ) -equivariant.

Definition 2.1.10.[ ncAn as noncommutative affine toric scheme] We shall call the above n construction the realization of ncA as a smooth noncommutative affine toric scheme over C, associated to the cone σ ⊂ NR.

By construction, the restriction of the toric scheme structure on ncAn to the master commutative subscheme An ,→ ncAn recovers the ordinary realization of An as a toric variety.

2.2 Soft noncommutative toric schemes associated to a fan n Recall the lattice N ' Z , with a ﬁxed basis (e1, ··· , en), the dual lattice M = Hom Z(N, Z), ∗ ∗ the evaluation pairing denoted by h , i : M × N → Z, the dual basis (e1, ··· , en) of M, and the associated R-vector spaces NR := N ⊗Z R and MR := M ⊗Z R. Let ∆ be a fan in N. (That is, ∆ is a set of rational strongly convex polyhedral cones σ in

NR such that (1) Each face of a cone in ∆ is also a cone in ∆; (2) The intersection of two cones in ∆ is a face of each; cf. [Fu: Sec. 1.4].) For σ, τ ∈ ∆, denote τ 4 σ or σ < τ if τ is a face of σ; and denote τ ≺ σ or σ τ if τ is a face of σ and τ 6= σ.

Deﬁnition 2.2.1.[ ∆-system, inverse ∆-system of objects in a category] Let C be a category, with the set of object denoted Object C and the set of morphisms of objects denoted Morphism C. Then a ∆-system (resp. inverse ∆-system) of objects in C is a correspondence F : ∆ → Object C and a choice of morphisms hτσ : F (τ) → F (σ) (resp. hστ : F (σ) → F (τ)) for all pairs of (σ, τ) ∈ ∆ × ∆ with τ ≺ σ such that hτσ ◦ hρτ = hρσ (resp. hτρ ◦ hστ = hσρ) for all ρ ≺ τ ≺ σ.

Recall how a variety Y (∆) can be associated to ∆, (e.g. [Fu]) as an inverse ∆-system of monoid algebras or a ∆-system of aﬃne schemes: Associated to each cone σ ∈ ∆ is a commutative monoid ∨ Mσ := σ ∩ M = {u ∈ M | hu, vi ≥ 0 for all v ∈ σ} ,.

By construction, there is a built-in monoid inclusion Mσ ,→ Mτ for τ ≺ σ. This gives rise to a ∆-system of aﬃne schemes { Uσ := Spec (C[Mσ]) }σ∈∆ ,

8 where C[Mσ] is the monoid algebra over C determined by Mσ, that glue to a toric variety Y (∆) through the system of inclusions of Zariski open sets

ιτσ : Uτ ,→ Uσ , for all τ ≺ σ ∈ ∆ , induced from the monoid inclusion Mσ ,→ Mτ , that by construction satisfy the gluing conditions:

ιτσ ◦ ιρτ = ιρσ , for all ρ ≺ τ ≺ σ ∈ ∆ .

× n The built-in (C ) -action on Uσ and the built-in torus embedding

n n T := TC := Spec (C[M]) ,→ Uσ × n n for each σ ∈ ∆ glues to a (C ) -action on Y (∆) and a torus embedding T ,→ Y (∆). In this section, we shall construct a class of noncommutative spaces, named soft noncommutative toric schemes, associated to ∆ by constructing an inverse ∆-system {M˘ σ}σ∈∆ of submonoids ˘ ∗ ∗−1 ∗ ∗ −1 in M := he1, e1 , ··· , en, en i via lifting the above construction to the edge-path monoids of commutative monoids with generators and propose soft gluings to bypass the generally unre- solvable issue of localizations of noncommutative rings when trying to gluing.

Assumption 2.2.2. [on fan ∆] Recall Sec. 2.1, theme: ncAn as a smooth noncommutative aﬃne toric scheme. To ensure that we have at least one uncomplicated chart in the atlas to begin with for the noncommutative toric scheme to be constructed, we shall assume that

· The cone generated by e1, ··· , en is in ∆. The corresponding chart ' ncAn is called the reference toric chart, cf. Deﬁnition 2.2.13. Fur- thermore, to ensure that we have enough good fundamental charts in the atlas before gluings, we shall assume in addition that

· All the maximal cones in ∆ are n-dimensional, simplicial, and of index 1.

In particular, Y (∆) is smooth.

Remark 2.2.3. [other noncommutative toric variety ] Based on various aspects of toric varieties and the language used, there are other notions and constructions of “noncommutative toric varieties”; for example, the work [K-L-M-V1], [K-L-M-V2] of Ludmil Katzarkov, Ernesto Lupercio, Laurent Meersseman, and Alberto Verjovsky. It is not clear if there is any connection among them. See Introduction for our motivation from D-branes in string theory.

Inverse ∆-systems of ∆-admissible submonoids of M˘

˘ ∗ ∗−1 ∗ ∗ −1 Recall from Example 1.7 the edge-path monoid M := he1, e1 , ··· , en, en i (cf. slightly different notation here) and the built-in monoid epimorphisms π : M˘ → M, π : C · M˘ → C · M and monoid-algebra epimorphism π : ChM˘ i → C[M] via commutatization.

Definition 2.2.4. [submonoid of M˘ admissible to cone in ∆] Let σ ∈ ∆. Then a submonoid M˘ 0 of M˘ is called admissible to σ if it satisfies: (0) M˘ is finitely generated. (1) The 0 monoid epimorphism π : M˘ → M induces a monoid epimorphism M˘ → Mσ by restriction. (2) 0 0 Anm ˘ ∈ M˘ is invertible in M˘ if π(m ˘ ) is invertible in Mσ.

9 Theorem 2.2.5. [existence of inverse ∆-system of admissible submonoids of M˘ ] There exists an inverse ∆-system {M˘ σ}σ∈∆ of submonoids of M˘ such that M˘ σ is admissible to σ for all σ ∈ ∆. We shall call such a system an inverse ∆-system of admissible submonoids of M˘ .

Proof. We proceed in three steps.

Step (a): Submonoids of M˘ admissible to maximal cones in ∆ Let σ be a maximal cone in ∆. By assumption, σ is n-dimensional, simplicial, and of index 1. ∨ Thus, σ is n-dimensional, simplicial, of index 1 in MR and the generators u1, ··· , un of the monoid σ∨ ∩ M generates M as well.

−1 Lemma 2.2.6. [submonoid admissible to maximal cone] Let u˘i ∈ π (ui) ⊂ M˘ , for i = 1, . . . , n. Then the submonoid hu,˘ ··· , u˘ni ⊂ M˘ is monoid-isomorphic to the free associative monoid of n letters hz1, ··· , zni.

∗ ∗ Proof. Recall the basis (e1, ··· , en) for M. Since (u1, ··· , un) generates M as well, up to Pn ∗ a relabelling of indices one may assume that ui = j=1 aijej with the coeﬃcient aii 6= 0, for i = 1 . . . , n. Since (˘u1, ··· , u˘n) projects to (u1, ··· , un) under π, as a word in 2n letters ∗ ∗−1 ∗ ∗ −1 0 ∗ ∗−1 e1, e1 , ··· , en, en ,u ˘i must contain a letter zi := ei , if aii > 0, or ei , if aii < 0. The 0 ∼ 0 0 ∼ correspondencesu ˘i 7→ zi 7→ zi induce monoid-isomorphisms hu˘1, ··· , u˘ni −→ hz1, ··· , zni −→ hz1, ··· , zni.

By construction, hu˘1, ··· , u˘ni is admissible to the maximal cone σ and we may take M˘ σ = hu˘1, ··· , u˘ni.

Step (b): Submonoids of M˘ admissible to lower-dimensional cones in ∆

For each maximal cone σ ∈ ∆, ﬁx a submonoid M˘ σ ⊂ M˘ admissible to σ as constructed in Step (a). For a cone τ ∈ ∆ of dimension < n, let ∆τ := {σ | τ ≺ σ, σ maximal cone} ⊂ ∆ and consider the submonoid of M˘ generated by all M˘ σ, where σ is a maximal cone in ∆ that contains τ: ˘ 0 ˘ ˘ Mτ := hMσ | σ ∈ ∆τ i ⊂ M. ∨ Since each σ ∈ ∆τ is strongly convex, the submonoid Mτ := τ ∩ M of M is generated by Mγ, γ ∈ ∆τ . It follows that the built-in monoid epimorphism restricts to a monoid epimorphism ˘ 0 ⊥ π : Mτ → Mτ . The submonoid of Mτ that consists of invertible elements in Mτ is τ ∩ M.

−1 ⊥ ˘ 0 ˘ 0 Lemma 2.2.7. [ﬁnite generatedness] The submonoid π (τ ∩ M) ∩ Mτ of Mτ is ﬁnitely generated.

Proof. For σ a maximal cone in ∆, let uσ,1, ··· , uσ,n be the generators of Mσ andu ˘σ,1, ··· , u˘σ,n be the generators of the monoid M˘ σ with π(˘uσ,i) = uσ,i. Then, by construction, ˘ ˘ 0 ˘ ⊥ ∨ {u˘σ,i | σ ∈ ∆τ , i = 1, . . . , n} ⊂ M generates the submonoid Mτ of M. Since τ ⊂ ∂τ := ∪ρ≺τ ρ and τ ∨ = P σ∨ is convex with τ ⊥ a linear stratum of ∂τ ∨, if m , m ∈ M satisfy the σ∈∆τ 1 2 τ ⊥ ⊥ condition m1 + m2 ∈ τ ∩ M, then both m1 and m2 must be in τ ∩ M. If follows that the −1 ⊥ ˘ 0 ˘ 0 submonoid π (τ ∩ M) ∩ Mτ of Mτ is generated by

⊥ {u˘σ,i | σ ∈ ∆τ , i = 1, . . . , n ; uσ,i ∈ τ } .

10 Let ˘ −1 ⊥ Sτ := hu˘σ,i , u˘σ,i | σ ∈ ∆τ , i = 1, . . . , n ; uσ,i ∈ τ i ˘ 0 and augment Mτ to ˘ ˘ 0 ˘ ˘ Mτ := hMτ , Sτ i ⊂ M.

Then, M˘ τ is now a submonoid of M˘ admissible to τ ∈ ∆. By construction, note that the ˘ ∗ ˘ ˘ submonoid Mτ of Mτ that consists of all the invertible elements of the monoid Mτ coincides with S˘τ ; and that M˘ 0 = M˘ .

Step (c). The inverse ∆-system {M˘ σ}σ∈∆ Consider the category of submonoids of M˘ with morphisms of objects given by inclusions of submonoids of M˘ and the collection {M˘ σ}σ∈∆ of submonoids of M˘ that are admissible to cones in ∆ constructed in Steps (a) and (b). For τ ≺ σ, ∆τ ⊃ ∆σ and hence M˘ σ ⊂ M˘ τ . For ρ ≺ τ ≺ σ, M˘ σ ⊂ M˘ τ ⊂ M˘ ρ naturally. This shows that the collection {M˘ σ}σ∈∆ is an inverse ∆-system of submonoids of M˘ and concludes the proof of the theorem.

˘ ˘ 0 Deﬁnition 2.2.8. [augmentation and diminishment] Let {Mσ}σ∈∆ and {Mσ}σ∈∆ be two ˘ ˘ ˘ 0 inverse ∆-systems of ∆-admissible submonoids of M such that Mσ ⊂ Mσ for all σ ∈ ∆. Then we ˘ ˘ 0 ˘ 0 ˘ say that {Mσ}σ∈∆ is a diminishment of {Mσ}σ∈∆ and {Mσ}σ∈∆ is an augmentation of {Mσ}σ∈∆. ˘ ˘ 0 ˘ 0 ˘ In notation, {Mσ}σ∈∆ 4 {Mσ}σ∈∆ and {Mσ}σ∈∆ < {Mσ}σ∈∆.

Similar arguments to the proof of Theorem 2.2.5 give the following:

Proposition 2.2.9. [completion to inverse ∆-system] Let ∆(n) be the set of n-dimensional ˘ 0 ˘ 0 ˘ cones in ∆. Given {Mσ}σ∈∆(n) such that Mσ is a submonoid of M admissible to σ, there ˘ ˘ ˘ ˘ 0 exists an inverse ∆-system {Mτ }τ∈∆ of admissible submonoids of M such that Mσ = Mσ for ˘ ˘ 0 σ ∈ ∆(n). We shall call {Mτ }τ∈∆ a completion of {Mσ}σ∈∆(n) to an inverse ∆-system of admissible submonoids of M˘ .

Proof. The same proof as the proof of Theorem 2.2.5 goes through as long as M˘ σ, σ ∈ ∆(n), in Step (a) is admissible to σ.

Proposition 2.2.10. [augmentation by ∆-indexed submonoids] Given an inverse ∆- ˘ 0 ˘ ˘ system {Mσ}σ∈∆ of admissible submonoids of M and a collection {Sσ}σ∈∆ of finitely generated ˘ ˘ ˘ ˘ 0 submonoids of M such that π(Sσ) ⊂ Mσ, there exists an augmentation {Mσ}σ∈∆ of {Mσ}σ∈∆ such that S˘σ ⊂ M˘ σ for all σ ∈ ∆. ∨ Proof. We begin with maximal cones σ ∈ ∆(n) and define M˘ σ := hM˘ σ, S˘σi. Since σ in this case ˘ is strongly convex in MR, Mσ must already be admissible to σ. We then proceed by induction by the condimension of cones in ∆ and assume that M˘ σ for σ ∈ ∆(i), 1 < k ≤ i ≤ n, are ˘ ˘ ˘ 0 ˘ ˘ ˘ constructed such that (1) Mσ is admissible to σ, (2) Mσ ⊃ hMσ, Sσi, (3) Mτ ⊃ Mσ if τ ≺ σ, for σ, τ ∈ ∆(i), 1 < k ≤ i ≤ n. Let τ ∈ ∆(k − 1) and ˘ 00 ˘ 0 ˘ ˘ Mτ := hMτ , Sτ , Mσ | σ ∈ ∆τ ∩ ∆(k)i . ˘ 00 −1 ⊥ Then Mτ ∩ π (τ ) is finitely generated and hence so is the submonoid ˘0 ˘ −1 ˘ 00 −1 ⊥ Sτ := {s ∈ M | s or s is in Mτ ∩ π (τ )}

11 of M˘ . Deﬁne ˘ ˘ 00 ˘0 Mτ := hMτ , Sτ i . ˘ ˘ ˘ 0 ˘ ˘ ˘ Then (1) Mτ is admissible to τ, (2) Mτ ⊃ hMτ , Sτ i, (3) Mτ ⊃ Mσ if τ ≺ σ for all σ ∈ ∆. Finally ˘ ˘ ˘ ˘ 0 we set M0 = M. By construction, {Mσ}σ∈∆ is an augmentation of {Mσ}σ∈∆ that satisﬁes S˘σ ⊂ M˘ σ for all σ ∈ ∆.

Corollary 2.2.11. [from ∆-indexed to inverse ∆-system] Let {S˘σ}σ∈∆ be a collection of ﬁnitely generated submonoids of M˘ such that π(S˘σ) ⊂ Mσ. Then there exists an inverse ∆-system {M˘ σ}σ∈∆ of admissible submonoids of M˘ such that S˘σ ⊂ M˘ σ for all σ ∈ ∆. Proof. Choose any inverse ∆-system of admissible submonoids of M˘ from Theorem 2.2.5 and apply Proposition 2.2.10.

Soft nocommutative toric schemes associated to ∆

Deﬁnition 2.2.12. [soft noncommutative toric schemes associated to ∆] (Continu- ing the previous notations.) Let {M˘ σ}σ∈∆ be an inverse ∆-system of admissible submonoids of M˘ such that M˘ σ ' hz1, ··· , zni for all maximal cones σ ∈ ∆(n) and M˘ 0 = M˘ and ˘ ˘ {Rσ}σ∈∆ := {ChMσi}σ∈∆ be the associated inverse ∆-system of monoid algebras over C. Then the corresponding ∆-system of noncommutative aﬃne schemes

Y˘ (∆) := U˘(∆) := {U˘σ}σ∈∆ is called a soft noncommutative toric scheme with smooth fundamental charts associated to ∆. Here, U˘σ is the noncommutative affine scheme associated to R˘σ (cf. Definition 2.1.1) for σ ∈ ∆, the notation U˘(∆) emphasizes that this is a gluing system of some particular charts, and the notation Y (∆) emphasizes that this is treated as a noncommutative space as a whole.1 With an abuse of language, we shall call Y˘ (∆) n-dimensional since the function ring of each fundamental chart is freely generated by n coordinate functions. ˘ For σ ∈ ∆, the monoid algebra ChMσi is called interchangeably the function ring or the local coordinate rings of the chart U˘σ. The assignment ˘ ˘ ˘ Uσ 7−→ Rσ := ChMσi , σ ∈ ∆ , ˘ is called the structure sheaf of Y (∆) and is denoted OY˘ (∆) as in ordinary Commutative Al- gebraic Geometry. By construction, for cones τ ≺ σ in ∆, one has a dominant morphism of 2 noncommutative affine schemes U˘τ → U˘σ and a monoid-algebra monomorphism

] ˘ ˘ ˘ ˘ ιτσ : OY˘ (∆)(Uσ) = ChMσi ,→ OY˘ (∆)(Uτ ) = ChMτ i .

1Here, we use the notation Y (∆), rather than X(∆) following [Fu], due to that such a space will be used as a target space for D-branes, whose world-volume is generally denoted XAz. 2Though our focus in this work is on the rings and their homomorphisms for Noncommutative Algebraic Geometry, we try to keep the contravariant underlying geometry as in Commutative Algebraic Geometry in mind ] whenever possible. This is why we use the notation ιτσ here and reserve ιτσ for the corresponding morphism of underlying “spaces”. Here, there is no indication that ιτσ : U˘τ → U˘σ in any sense is an inclusion, though it is true that, when restricted to Y (∆), ιτσ : Uτ → Uσ is an open set inclusion in the sense of aﬃne schemes in Commutative Algebraic Geometry.

12 These replace the role of ‘open sets’ and ‘restriction to open sets’ respectively in the usual deﬁnition of the ‘structure sheaf ’ of a scheme in Commutative Algebraic Geometry. An ideal sheaf I˘ on Y˘ is an inverse ∆-system {I˘σ}σ∈∆, where I˘σ is a two-sided ideal of ] R˘σ, such that ιτσ(Iσ) = Iτ . Given an ideal sheaf I˘ = {I˘σ}σ∈∆ on Y˘ (∆), one can form an ˘ ˘ ] ˘ ˘ ˘ ˘ inverse ∆-system of C-algebras {Rσ/Iσ}σ∈∆, with the inclusion ιτσ : Rσ/Iσ ,→ Rτ /Iτ , for ] τ ≺ σ ∈ ∆, naturally induced from ιτσ. We will think of this inverse ∆-system as deﬁning a soft ˘ ˘ noncommutative closed subscheme Z of Y (∆) and denoted its structure sheaf OZ˘ and write the ˘ quotient as OY˘ (∆) → OZ˘. Redenote I by IZ˘. Then one has a short exact sequence

0 −→ IZ˘ −→ OY˘ (∆) −→ OZ˘ −→ 0 .

0 n0 0 0 0 0 Let N be another lattice (isomorphic to Z for some n via a fixed basis (e1, ··· , en0 ) of N ), ∆0 be a fan in N 0 that satisfies Assumption 2.2.2, and Y˘ (∆0) be a soft noncommutative toric R scheme associated to ∆0, with the underlying inverse ∆0-system of submonoids of M˘ 0 denoted ˘ 0 {Mσ0 }σ0∈∆0 .A toric morphism ϕ˘ : Y˘ (∆) −→ Y˘ (∆0) is a homomorphism ϕ : N → N 0 of lattices that satisfies the conditions: (1) ϕ induces a map between fans (in notation, ϕ : ∆ → ∆0); i.e., for each σ ∈ ∆ there exists a σ0 ∈ ∆0 such that ϕ(σ) ⊂ σ0. (2) The induced monoid-homomorphismϕ ˘] : M˘ 0 → M˘ restricts to a monoid homomorphism ] ˘ 0 ˘ 0 ˘ 0 ˘ ϕ˘σσ0 : Mσ0 → Mσ whenever ϕ(σ) ⊂ σ . (Denote the induced ChMσ0 i → ChMσi also by ] ϕ˘σσ0 .) (Cf. Footnote 2.) Thus, a toric morphismϕ ˘ : Y˘ (∆) → Y˘ (∆0) is a map of lattices ϕ : N → N 0 that induces a 0 ] ˘ 0 ˘ (∆ , ∆)-system of monoid-algebra homomorphisms {ϕ˘σσ0 : ChMσ0 i → ChMσi}σ0∈∆0, σ∈∆, ϕ(σ)⊂σ0 . The nature of this collection justifies it be denoted

] ϕ˘ : OY˘ (∆0) −→ OY˘ (∆) .

Deﬁnition 2.2.13. [reference toric chart and fundamental toric charts of Y˘ (∆)] (Con- tinuing Deﬁnition 2.2.12.) A chart U˘σ of Y˘ (∆) that is associated to a maximal cone σ ∈ ∆(n) is called a fundamental toric chart of Y˘ (∆). Among them, the one associated to the cone generated by the given basis (e1, ··· , en) of the lattiice N is called the reference toric chart of Y˘ (∆).

Fundamental toric charts are all isomorphic to the smooth noncommutative affine n-space ncAn. These charts should be thought of as the basic, good pieces to be glued via the sub- ˘ collection {Uτ }τ∈∆−∆(n). The reference chart ensures that we have at least one fundamental toric chart U˘σ such that the morphism U˘0 → U˘σ of noncommutative affine schemes resembles the inclusion of an Zariski open set. Together, this allows one to think of Y˘ (∆) as a “partial compactification” of ncAn via soft gluings of additional copies of ncAn’s.

Deﬁnition 2.2.14. [softening of noncommutative toric scheme] Let Y˘ (∆) and Y˘ 0(∆) be two soft noncommutative toric schemes associated to a fan ∆, with the underlying inverse ˘ ˘ 0 ˘ 0 ∆-system of monoid algebras {ChMσi}σ∈∆ and {ChMσi}σ∈∆ respectively. Then Y (∆) is called ˘ ˘ ˘ 0 ˘ ˘ 0 a softening of Y (∆) if Mσ = Mσ for maximal cones σ ∈ ∆ (i.e. Y (∆) and Y (∆) have identical ˘ ˘ 0 ˘ 0 fundamental toric charts) and Mσ ⊂ Mσ for nonmaximal cones σ ∈ ∆ (i.e. the gluings in Y (∆) is softer than those in Y˘ (∆)). By construction, there is a built-in toric morphism ϕ˘ : Y˘ 0(∆) → Y˘ (∆), called the softening morphism.

13 Remark 2.2.15. [meaning of softening: comparison to Isom -functor for moduli stack] Introduc- ing the notion of ‘softening’ in the play is meant to remedy the fact that we don’t have a good theory of localizations of a noncommutative ring to apply to our construction of noncommutative schemes. Rather we keep a collection of charts as principal ones and the remaining charts serve for the purpose of gluing these principal charts. These ‘secondary charts’ are then allowed to be ‘dynamically adjusted’ to fulﬁll the purpose of serving as a medium between principal charts. One may compare this to the gluing in the construction of a moduli stack. There one cannot directly glue two charts for the moduli stack as schemes but rather has to create the medium chart via the Isom -functor, cf. [Mu]. See Sec. 3 how softening is used to construct invertible sheaves and twisted sections on a soft noncommutative toric scheme.

n (c) The T -action n The T -action on C[M] leaves each C · m, m ∈ M, invariant. As C = Center (ChM˘ i), the n n T -action on C[M] naturally lifts to a T -action on ChM˘ i:

t ChM˘ i / ChM˘ i

π π

t n C[M] / C[M] , t ∈ T ,

˘ n ˘ that leaves each C·m˘ ,m ˘ ∈ M, invariant. It follows that T leaves each ChMσi, σ ∈ ∆, invariant ] ˘ ˘ n n ˘ and that ιτσ : ChMσi ,→ ChMτ i, τ ≺ σ, are T -equivariant. This deﬁnes the T -action on Y (∆).

(d) The noncommutative toroidal morphism from the inclusion of the 0-cone 0 ≺ σ ∈ ∆ ˘ ˘ n Since 0 4 σ, for all σ ∈ ∆, and M0 = ChMi, one has a built-in T -equivariant morphism n T˘ := Scheme (ChM˘ i) −→ Y˘ (∆) .

n Which plays the role of the toroidal embedding T ,→ Y (∆) in the commutative case.

n (e) The built-in T -equivariant embedding Y (∆) ,→ Y˘ (∆)

The monoid epimorphism π : M˘ → M restricts to monoid epimorphisms π : M˘ σ → Mσ, for n all σ ∈ ∆. Which in turn gives a ∆-collection of T -equivariant monoid-algebra epimorphisms ˘ {ChMσi →→ C[Mσ]}σ∈∆ that commute with inclusions:

˘ ˘ ChMσi / ChMτ i

π π C[Mσ] / C[Mτ ] , τ ≺ σ ∈ ∆ .

n This deﬁnes a built-in T -equivariant embedding Y (∆) ,→ Y˘ (∆).

(f) The master commutative subscheme of Y˘ (∆) ˘ ˘ The inclusions ChMσi ,→ ChMτ i, τ ≺ σ ∈ ∆, induce C-algebra homomorphisms ] ˘ ˘ ˘ ˘ ˘ ˘ τσ : ChMσi/[ChMσi, ChMσi] −→ ChMiτ /[ChMτ i, ChMτ i] , τ ≺ σ ∈ ∆ .

This deﬁnes another inverse ∆-system of commutative C-algebras ˘ ˘ ˘ ] ({ChMσi/[ChMσi, ChMσi]}σ∈∆, {τσ}τ≺σ∈∆)

14 and hence a (generally singular) commutative scheme Y˘ (∆)♣ over C. By construction any morphism from a commutative scheme X to Y˘ factors through a morphism X → Y˘ (∆)♣ ,→ Y˘ (∆). In particular, one has commutative diagrams

] ˘ ˘ ˘ τσ ˘ ˘ ˘ ChMσi/[ChMσi, ChMσi] / ChMiτ /[ChMτ i, ChMτ i]

C[Mσ] / C[Mτ ] , τ ≺ σ ∈ ∆ , and hence Y (∆) ⊂ Y˘ (∆)♣ ⊂ Y˘ (∆). Note that, by construction, ˘ ˘ ˘ ∼ ChMσi/[ChMσi, ChMσi] −→−→ C[Mσ] for σ ∈ ∆(n) ∪ {0}. However, such isomorphism on local charts may not holds for τ ∈ ∆(k), 1 ≤ k ≤ n − 1.

Deﬁnition 2.2.16. [master commutative subscheme of Y˘ (∆)] Y˘ (∆)♣ is called the master commutative subscheme of Y˘ (∆).

3 Invertible sheaves on a soft noncommutative toric scheme, twisted sections, and soft noncommutative schemes via toric geometry

We construct in this section a class of noncommutative spaces, named ‘soft noncommutative schemes’ from invertible sheaves and their twisted sections on Y˘ (∆).

Modules and invertible sheaves on a soft noncommutative toric scheme

˘ Definition 3.1. [sheaf on Y (∆) and OY˘ (∆)-module] (1) An inverse ∆-system of objects in a category C (cf. Definition 2.2.1) is also called a sheaf (of objects in C) on Y˘ (∆). For example, ˘ ˘ OY˘ (∆) is a sheaf of C-algebras on Y (∆). (Which justifies its name: the structure sheaf of Y (∆)).

(2) Let Y˘ (∆) = {U˘σ}σ∈∆ be a soft noncommutative toric scheme with the underlying inverse ˘ ] ∆-system of monoid algebras ({Rσ}σ∈∆, {ιτσ}τ, σ∈∆, τ≺σ). A left OY˘ (∆)-module is an inverse ∆- system F˘ := ({F˘σ}σ∈∆, {hστ }τ, σ∈∆, τ≺σ) , where F˘ is a left R˘ -module and h : ι] F˘ := R˘ ⊗ F˘ → F˘ on U˘ such that the following σ σ στ στ σ τ R˘σ σ τ τ diagram commutes hσρ

] ] ιτρhστ ] hτρ $ ισρF˘σ / ιτρF˘τ / F˘ρ on U˘ρ for ρ ≺ τ ≺ σ. An element of F˘σ is called a local section of F˘ over U˘σ.A section (or global section)of F˘ is a collection {s˘σ}σ∈∆ of local sectionss ˘σ ∈ F˘σ, σ ∈ ∆, of F˘ such that hστ (˘sσ) =s ˘τ , τ ≺ σ ∈ ∆. In the last expression,s ˘ is identiﬁed as 1 ⊗ s˘ ∈ R˘ ⊗ F˘ . The -vector space of sections of σ σ τ R˘σ σ C F˘ on Y˘ (∆) is denoted H0(Y˘ (∆), F˘) or simply H0(F˘).

15 A homomorphism

˘ ˘ ˘0 ˘0 0 f : F := ({Fσ}σ∈∆, {hστ }τ, σ∈∆, τ≺σ) −→ F := ({Fσ}σ∈∆, {hστ }τ, σ∈∆, τ≺σ)

˘ ˘0 of left OY˘ (∆)-modules is a collection {fσ}σ∈∆, where fσ : Fσ → Fσ is a homomorphism of left R˘σ-modules, such that the following diagrams commute

] hστ ιστ F˘σ / F˘τ

] ιστ (fσ) fτ 0 ] ˘0 hστ ˘0 ιστ Fσ / Fτ for τ ≺ σ ∈ ∆. f is called injective (i.e. f a monomorphism) if in addition fσ is injective for all σ ∈ ∆; surjective (i.e. f an epimorphism) if in addition fσ is surjective for all σ ∈ ∆; an isomorphism if in addition fσ is an isomorphism of left R˘σ-modules for all σ ∈ ∆. n ˘ ˘ n (3) Recall the built-in T -action g• on Y (∆). Then F is called T -equivariant if there exist ∗ ˘ ∼ ˘ n isomorphisms φt : gt F −→ F, t ∈ T , of OY˘ (∆)-modules such that φt2 ◦ φt1 = φt1+t2 :

φt1+t2

φt φt g∗ (g∗ F˘) 1 g∗ F˘ 2 $ F˘ t2 t1 / t2 /

n n ˘ ˘ for all t1, t2 ∈ T . Explicitly, note that the T -action on Y (∆) leaves each chart Uσ, σ ∈ ∆ invariant and φt restricts to an R˘σ-module homomorphism ˘t ˘ φt : Fσ −→ Fσ , withs ˘ 7−→ φt(˘s) =: gt · s˘ ˘ ˘t ∗ ˘ ˘ on each chart Uσ, σ ∈ ∆. Here Fσ := gt Fσ, which is the same C-vector space as Fσ but with the ˘ t ] n ˘ Rσ-module structure defined byr ˘ · s˘ := gt(˘r) · s˘. This defines a T -action on Fσ that satisfies ] n ˘ ˘ n ˘ φt(gt(˘r) · s˘) =r ˘ · (gt · s˘) by tautology, for t ∈ T ,r ˘ ∈ Rσ, ands ˘ ∈ Fσ. The T -action on Fσ, σ ∈ ∆, is equivariant under gluings:

hστ (gt · s˘) = gt · hστ (˘s) ,

˘ n fors ˘ ∈ Fσ, t ∈ T , and τ ≺ σ ∈ ∆.

(4) Similarly, for right and two-sided OY˘ (∆)-modules. ˘ ˘ ] ˘ ˘ F˘ (5) Let A = ({Aσ}σ∈∆, {ιτσ}τ≺σ∈∆) be a sheaf of C-algebras, F = ({Fσ}σ∈∆, {hστ }τ≺σ∈∆) ˘ ˘ ˘ G˘ ˘ ˘ a right A-module, and G = ({Gσ}σ∈∆, {hστ }τ≺σ∈∆) a left A-module, all three on Y (∆). Then ˘ ˘ ˘ ˘ ˘ ˘ ˘ deﬁne the tensor product F ⊗A˘ G of F and G over A to be the following (right-on-F, left-on-G) A˘-module on Y˘ (∆)

˘ ˘ F˘ ⊗ G˘ := ({F˘ ⊗ G˘ } , {hF ⊗ hG } ) . A˘ σ A˘σ σ σ∈∆ στ στ τ≺σ∈∆

(6) Let ˘ ˘ ˘ 0 ˘ ϕ˘ : Y (∆) = {Rσ}σ∈∆ −→ Y (∆ ) = {Sσ0∈∆0 }σ0∈∆0 ˘ ˘ F˘ be a toric morphism (cf. Deﬁnition 2.2.12), F := ({Fσ}σ∈∆, {hστ }τ≺σ∈∆) a left OY˘ (∆)-module ˘ ˘ ˘ G˘ ˘ 0 on Y (∆), and G := ({Gσ0 }σ0∈∆0 , {hσ0τ 0 }τ 0≺σ0∈∆0 ) a left OY˘ (∆0)-module on Y (∆ ). Recall the

16 underlying homomorphism N → N 0 of lattices and the induced -linear map N → N 0 , both R R R denoted still by ϕ, (cf. Definition 2.2.12). For each σ0 ∈ ∆0, 0 ∆σ := {σ ∈ ∆ | ϕ(σ) ⊂ σ0} ˘ is a subfan of ∆ and specifies a subsystem OY˘ (∆σ0 ) := {Rσ}σ∈∆σ0 of OY˘ (∆). Through the ] ˘ 0 ˘ σ0 restriction ofϕ ˘ : OY˘ (∆0) → OY˘ (∆) to Sσ0 → OY˘ (∆0), the C-vector space H (Y (∆ ), F|Y˘ (∆σ0 )) is ˘ F˘ ˘ rendered a left Sσ0 -module. The gluing data {hστ }τ≺σ∈∆ of F gives rise to a gluing data for the 0 0 ˘ σ0 0 ∆ -collection {H (Y (∆ ), F|Y˘ (∆σ0 ))}σ0∈∆0 and turn the ∆ -collection into a left OY˘ (∆0)-module, denotedϕ ˘∗F˘ and named the direct image sheaf or pushforward of F˘ underϕ ˘. 0 0 0 0 0 For each σ ∈ ∆, there exists a unique ρσ ∈ ∆ such that ϕ(σ) ⊂ ρσ and that ρσ 4 σ 0 0 0 0 0 for all σ ∈ ∆ with ϕ(σ) ⊂ σ . The property ρτ 4 ρσ for τ ≺ σ ∈ ∆ and the gluing data G˘ 0 0 0 ˘ ˘ 0 ˘ {hσ0τ 0 }τ ≺σ ∈∆ of G render the ∆-collection {Gρσ }σ∈∆ a sheaf of C-vector spaces on Y (∆), −1 ˘ ˘ −1 denotedϕ ˘ G and named the inverse image sheaf of G underϕ ˘. In particular,ϕ ˘ OY˘ (∆0) is ˘ −1 ˘ −1 a sheaf of C-algebras on Y (∆) and, by construction,ϕ ˘ G is a leftϕ ˘ OY˘ (∆0)-module. Since ] −1 ϕ˘ : OY˘ (∆0) → OY˘ (∆) renders OY˘ (∆) a two-sidedϕ ˘ OY˘ (∆0)-module, one can define further a ∗ −1 left O ˘ -moduleϕ ˘ G˘ := O ˘ ⊗ −1 ϕ˘ G on Y˘ (∆), named the pullback of G˘ underϕ ˘. Y (∆) Y (∆) ϕ˘ OY˘ (∆0) Similarly, for right modules and two-sded modules on Y˘ (∆) and Y˘ (∆0).

Example 3.2. [ideal sheaf and structure sheaf of closed subscheme] Recall Deﬁni- ˘ tion 2.2.12. Then, an ideal sheaf IZ˘ on Y (∆) and the associated structure sheaf OZ˘ of a soft ˘ ˘ noncommutative closed subscheme Z of Y (∆) are both two-sided OY˘ (∆)-modules.

˘ Deﬁnition 3.3. [invertible OY˘ (∆)-module/invertible sheaf/line bundle on Y (∆)] (Con- tinuing Deﬁnition 3.1.) A left (resp. right, two-sided ) invertible OY˘ (∆)-module (or invertible sheaf ˘ or line bundle) on Y (∆) is a left (resp. right, two-sided) OY˘ (∆)-module

L˘ = ({F˘σ}σ∈∆, {hστ }τ, σ∈∆, τ≺σ) such that F˘σ ' R˘σ as left (resp. right, two-sided) R˘σ-modules for σ ∈ ∆ and hστ is an isomorphism of left (resp. right, two-sided) R˘τ -modules for τ, σ ∈ ∆, τ ≺ σ.

Lemma 3.4. [invertible OY˘ (∆)-module: basic description] (Continuing the notation from ˘ ˘ Sec. 2.2.) Let L be a (left) invertible OY˘ (∆)-module. Then L can be described by a collection

⊥ × ˘ σ, τ ∈ ∆, τ ≺ σ, π(m ˘ στ ) ∈ τ ∩ M, {cστ m˘ στ }τ≺σ∈∆, • := cστ m˘ στ ∈ C Mτ . cσρm˘ σρ = cστ cτρm˘ στ m˘ τρ for ρ ≺ τ ≺ σ ˘ ˘ ˘ Here, m˘ σσ0 acts on Fσσ0 ' Rσσ0 as a left Rσσ0 -module homomorphism given by ‘the multiplication by m˘ σσ0 from the right’. In particular, m˘ τρ ◦ m˘ στ =m ˘ στ m˘ τρ. Two such collections 0 0 {cστ m˘ στ }τ≺σ∈∆, • and {cστ m˘ στ }τ≺σ∈∆, • deﬁne isomorphic invertible OY˘ (∆)-modules if and only if there exists a collection × ˘ ⊥ {cσm˘ σ}σ∈∆,• := {cσm˘ σ ∈ C Mσ | σ ∈ ∆, π(m ˘ σ) ∈ σ ∩ M} such that 0 −1 0 −1 cστ = cσ cστ cτ and m˘ στ =m ˘ σ m˘ στ m˘ τ .

17 Proof. Observe that the multiplicative monoid of invertible elements of the monoid algebra × ChM˘ i is given by C M˘ and that the multiplicative monoid of invertible elements of the monoid ˘ × ˘ × ˘ −1 ⊥ algebra Rσ, σ ∈ ∆, is contained in C M and is given by C Mσ ∩ π (σ ∩ M). It follows from Definition 3.3 that an invertible OY˘ (∆)-module is determined by gluings of the collection {R˘σ}σ∈∆ for each pair τ ≺ σ as R˘τ -modules. This gives the gluing data {cστ m˘ στ }τ≺σ∈∆,• in the Statement. Different choices of isomorphisms F˘σ ' R˘σ (i.e. local trivializations of L) define isomorphic invertible OY˘ (∆)-modules.

Proposition 3.5. [existence of invertible sheaf after passing to softening] Let Y˘ (∆) be an (n-dimensional) noncommutative toric scheme associated to a fan ∆. Recall the built-in ˘ inclusion Y (∆) ⊂ Y (∆) and let L be an invertible OY (∆)-module on Y (∆). Then, there exists a softening Y˘ 0(∆) → Y˘ (∆) of Y˘ (∆) such that L, now on Y (∆) ⊂ Y˘ 0(∆), extends to an invertible ˘ 0 OY˘ 0(∆)-module on Y (∆).

n Proof. Under Assumption 2.2.2, Y (∆) is smooth and thus L'OY (∆)(D) for some T -invariant Cartier divisor D on Y (∆) and can be specified by a collection {mσ ∈ M}σ∈∆(n), where mσ ∈ M −1 ⊥ and ∆(n) is the collection of maximal cones in ∆, that satisfies mσ mτ ∈ (σ ∩ τ) ∩ M, for σ, τ ∈ ∆(n). (Cf. [Fu: Sec. 3.4], with the monoid M here presented multiplicatively for convenience.) For each τ ∈ ∆(k), k ≤ n − 1, fix an σ ∈ ∆(n) such that τ ≺ σ and set mτ = mσ. −1 ⊥ This extend {mσ}σ∈∆(n) to a collection {mσ}σ∈∆ such that mσ mτ ∈ (τ ∩ σ) ∩ M. Let {M˘ σ}σ∈∆ be the underlying inverse ∆-system of monoids associated to Y˘ (∆) and, for each σ ∈ ∆, fix anm ˘ σ ∈ M˘ σ such that π(m ˘ σ) = mσ. Set

−1 {m˘ στ }τ≺σ∈∆ = {m˘ σ m˘ τ }τ≺σ∈∆ .

Then,

⊥ π(m ˘ στ ) ∈ τ ∩ M for τ ≺ σ ∈ ∆ , andm ˘ στ m˘ τρ =m ˘ σρ for ρ ≺ τ ≺ σ ∈ ∆ .

This is almost the gluing data in Lemma 3.4 for an invertible OY˘ (∆)-module except that in generalm ˘ στ ∈/ M˘ τ . To remedy this, let

S˘σ = the empty set for σ ∈ ∆(n) ,

S˘τ = hm˘ στ | τ ≺ σ ∈ ∆i for τ ∈ ∆(k), k ≤ n − 1 .

This defines a collection of ∆-indexed finitely generated submonoids of M˘ . It follows from ˘ ˘ 0 Proposition 2.2.10 that one can augment {Mσ}σ∈∆ to an inverse ∆-system {Mσ}σ∈∆ of sub- ˘ ˘ ˘ 0 ˘ monoids of M such that Sσ ⊂ Mσ. Furthermore, since Sσ = empty set for σ ∈ ∆(n), it follows ˘ 0 ˘ from the proof of Proposition 2.2.10 that it can be made that Mσ = Mσ for σ ∈ ∆(n). Thus, ˘ 0 ˘ 0 ˘ ˘ {M }σ∈∆ defines a softening Y (∆) of Y (∆) and L extends to an invertible OY˘ 0(∆)-module L on Y˘ 0(∆).

Since a soft noncommutative toric scheme associated to ∆ always exists (cf. Theorem 2.2.5, Proposition 2.2.9, Deﬁnition 2.2.12), Proposition 3.5 can be rephrased as:

18 Proposition 3.50. [existence of invertible sheaf] Let L be an invertible sheaf on the smooth toric variety Y (∆). Then there exists a soft noncommutative toric scheme Y˘ (∆) ⊃ Y (∆) associated to ∆ such that L extends to an invertible sheaf L˘ on Y˘ (∆).

Note that the pullback of an invertible sheaf under a morphism of noncommutative toric schemes is an invertible sheaf. In particular, any invertible sheaf on Y˘ (∆) pulls back to invertible sheaves on softenings of Y˘ (∆).

Twisted sections of an invertible sheaf on Y˘ (∆)

Let L be an invertible sheaf on the n-dimensional smooth toric variety Y (∆). Then L is isomor- P phic to a T-Cartier divisor OY (∆)(D) = OY (∆)( τ∈∆(1) aτ Dτ ), where ∆(1) ⊂ ∆ is the set of rays (i.e. 1-dimensional cones) in ∆, Dτ is the T-invariant Weil divisor on Y (∆) associated to τ ∈ ∆(1), and aτ ∈ Z. The divisor D determines an mσ ∈ M for each maximal cone σ ∈ ∆(n):

hmσ, vτ i = − aτ , for all τ ∈ ∆(1) such that τ ≺ σ .

Here vτ ∈ τ ∩ N is the ﬁrst lattice point on the ray τ. Writing the commutative monoid M multiplicatively as a multiplicatively closed subset of R0 = C[M] and passing to the isomorphism ∨ L'OY (∆)(D), then L(Uσ) = Rσ · mσ = C[σ ∩ M] · mσ, σ ∈ ∆(n). It follows that, if deﬁning the (possibly empty) polytope in MR associated to D

\ ∨ PD := σ · mσ = {u ∈ MR | hu, vτ i ≥ −aτ , for all τ ∈ ∆(1)} ⊂ MR, σ∈∆(n)

0 then the C-vector space H (Y (∆), L) of sections of L is realized as

0 M H (Y (∆), OY (∆)(D)) = C · u . u∈PD∩M (Cf. [Fu: Sec. 3.3 & Sec. 3.4].)

Deﬁnition 3.6. [twisted section of invertible sheaf] Let L˘ = ({F˘σ}σ∈∆, {hστ }τ≺σ∈∆) be a (left) invertible sheaf on a soft noncommutative toric scheme Y˘ (∆) = {R˘σ}σ∈∆. Two local sectionss ˘1, s˘2 ∈ F˘σ, σ ∈ ∆, are said to be equivalent, in notation,s ˘1 ∼ s˘2, if there exists an ˘ ∗ ˘ invertible elementr ˘ ∈ Rσ such thats ˘1 =r ˘s˘2. This is clearly an equivalence relation on Fσ, σ ∈ ∆. A twisted section of L˘ is a collection

s˘ = {s˘σ}σ∈∆ of local sectionss ˘σ ∈ F˘σ of L˘ such that

hστ (˘sσ) ∼ s˘τ , for τ ≺ σ ∈ ∆ .

Fix a local trivializing isomorphism F˘σ ' R˘σ of left R˘σ-modules, σ ∈ ∆. Then,s ˘ is specifoed uniquely by {r˘σ}σ∈∆, withr ˘ ∈ R˘σ. We will called {r˘σ}σ∈∆ the presentation ofs ˘ with respect to the local trivialization of L˘. Diﬀerent choices of local trivializations of L˘ give presentations ofs ˘ that diﬀer by a right multiplication of unit (i.e. invertible element) chart by chart.

Lemma 3.7. [twisted section under pullback] Let φ : Y˘ 0(∆) → Y˘ (∆) be a morphism of soft noncommutative toric schemes, L˘ be an invertible sheaf on Y˘ (∆), and s˘ be a twisted section of L˘. Then, φ∗s˘ is a twisted section of the pullback φ∗L˘ of L˘ to Y˘ 0(∆). In particular, a twisted section pulls back to a twisted section under softening.

19 Proof. This follows from the fact that φ] takes an invertible element to an invertible element chart by chart.

Proposition 3.8. [extension to twisted section after passing to softening] Let L˘ be an invertible sheaf on Y˘ (∆), L be the restriction of L˘ to Y (∆) ⊂ Y˘ (∆), and s1, ··· , sk ∈ H0(Y (∆), L) be sections of L on Y (∆). Then, there exists a softening Y˘ 0(∆) → Y˘ (∆) such that 0 0 all s1 ··· , sk of L extend to twisted sections of the pullback L˘ of L˘ to Y˘ (∆).

Proof. Present L as a T-Cartier divisor OY (∆)(D), recall the polytope PD in MR associated to D, and denote the section of L corresponding to m ∈ PD ∩ M by sm. Since each si is a C-linear combination of ﬁnitely many sm’s, m ∈ PD ∩ M, it follows from Lemma 3.7 that we only need 0 to prove the proposition for a section of L in the form sm0 for some m ∈ PD ∩ M. ˘ ˘ ˘ ˘ Write out more explicitly: Y (∆) = {ChMσi}σ∈∆ and L = ({ChMσi}σ∈∆, {cρτ m˘ στ }τ≺σ∈∆) under a local trivialization (cf. Lemma 3.4). Recall the collection {mσ}σ∈∆(n) ⊂ M associated to D from the beginning of the theme. For any τ ∈ ∆ − ∆(n), ﬁx an σ ∈ ∆(n) such that τ ≺ σ and assign mτ to be mσ. This enlarges the collection {mσ}σ∈∆(n) to a collection {mσ}σ∈∆ with the property that −1 ⊥ mσ mτ ∈ τ ∩ M, for τ ≺ σ ∈ ∆ . Here, we write the operation of the commutative monoid M multiplicatively. 0 ˘ Choosem ˘ σ, m˘ σ ∈ M such that 0 0 0 ˘ π(m ˘ σ) = mσ, π(m ˘ σ) = m , m˘ σ ∈ Mσ · m˘ σ,

0 for all σ ∈ ∆. Since π : M˘ → M restricts to a surjection M˘ σ · m˘ σ → Mσ · mσ and m ∈ Mσ · mσ, 0 0 0 ˘ ˘ such a choice always exists. It follows thatm ˘ σ =r ˘σ · m˘ σ for somer ˘σ ∈ Mσ ⊂ Rσ, σ ∈ ∆. Consider now the collection 0 0 s˘ := {r˘σ}σ∈∆ ˘ of local sections of L. By construction, it recovers the section sm0 of L when restricted to Y (∆) ⊂ Y˘ (∆). For τ ≺ σ ∈ ∆,

0 0 ˘ hστ (˘rσ) =r ˘σ · cστ m˘ στ ∈ Rτ .

As elements in M˘ , 0 0 0 −1 0 r˘σ m˘ στ = (˘rσ m˘ στ r˘τ ) · r˘τ . Since 0 0 −1 −1 ⊥ π(˘rσ m˘ στ r˘τ ) = (mσ mτ ) · π(m ˘ στ ) ∈ τ ∩ M, 0 0 −1 ˘ π(˘rσ m˘ στ r˘τ ) is invertible in C[Mτ ]. Thus,s ˘ is almost a twisted section of L except that 0 0 −1 ˘ the twisting factorr ˘σ m˘ στ r˘τ may not yet be in Mτ . The same argument as in the proof of Proposition 3.5 and applying Proposition 2.2.10 show that there exists a softening φ : Y˘ 0(∆) → Y˘ (∆) such that the pullback collection φ∗s˘0 is indeed a twisted section of the pullback invertible sheaf φ∗L˘ on Y˘ 0(∆). This proves the proposition.

Since a soft noncommutative toric scheme associated to ∆ with an invertible sheaf that extends L on Y (∆) exists (cf. Proposition 3.50), Proposition 3.8 is equivalent to:

20 Proposition 3.80. [extension to twisted section] Let L be an invertible sheaf on Y (∆) 0 and s1, ··· , sk ∈ H (Y (∆), L) be sections of L. Then, there exists a soft noncommutative toric scheme Y˘ (∆) with an invertible sheaf L˘ that restricts to L on Y (∆) ⊂ Y˘ (∆) such that si extends to a twisted section s˘i of L˘, for i = 1, . . . , k.

Soft noncommutative closed subschemes of Y˘ (∆) associated to twisted sections of invertible sheaves on Y˘ (∆)

Continuing the notation and discussion of the previous theme. Lets ˘i = {s˘i, σ}σ∈∆ be a (non- zero) twisted section of an invertible sheaf L˘i on Y˘ (∆), for i = 1, . . . , k. With respect to a local trivialization of each L˘i,s ˘i has a presentation {r˘i, σ}σ∈∆, withr ˘i, σ ∈ R˘σ, σ ∈ ∆. A different local trivialization of L˘i gives rise to a different presentation ofs ˘i of the form {r˘i, σu˘i, σ}σ∈∆, whereu ˘i, σ is a unit (i.e. invertible element) of R˘σ. Denote Γ = {s˘1, ··· , s˘k}. It follows that the two-sided ideal I˘Γ, σ := (˘r1, σ, ··· , r˘k, σ) of R˘σ, σ ∈ ∆, associated to Γ is independent of the local trivialization of L˘i, i = 1, . . . , k. This defines an ideal sheaf IΓ := {I˘Γ, σ}σ∈∆ on Y˘ (∆) and hence a soft noncommutative closed subscheme Z˘Γ of Y˘ (∆).

Deﬁnition 3.9. [soft noncommutative closed subscheme associated to twisted sections] Z˘Γ is called the soft noncommutative closed subscheme of Y˘ (∆) associated to Γ. When Γ = {s˘}, Z˘s˘ := Z˘Γ is called the soft noncommutative hypersurface of Y˘ (∆) associated tos ˘.

Deﬁnition 3.10. [soft noncommutative scheme/space (via toric geometry)] A soft noncommutative toric scheme Y˘ (∆) associated to a fan ∆ or a soft noncommutative closed subscheme Z˘ of a Y˘ (∆) for some ∆ will be called a soft noncommutative scheme via toric geometry, or simply a soft noncommutative scheme, or even a soft noncommutative space. The ˘ structure sheaf OY˘ of such a noncommutative space Y is deﬁned to the inverse ∆-system of C-algebras that describes Y˘ .

It is this class of noncommutative spaces that we will study further in sequels. Defini- tion 2.2.12 and Definition 3.1 can be generalized routinely to soft noncommutative schemes and their modules. A straightforward generalization of Proposition 3.80 to a finite collection of invertible sheaves with sections {(L1, s1), ··· , (Lk, sk)} on Y (∆) implies that:

Corollary 3.11. [commutative complete intersection subscheme] Any complete intersection subscheme Z of a smooth toric variety Y (∆) embeds as a commutative closed subscheme of a soft noncommutative scheme Z˘ in a Y˘ (∆) (⊃ Y (∆)) such that Z˘ ∩Y (∆) = Z. In particular, this applies to complete intersection Calabi-Yau spaces in a smooth toric variety.

Indeed it is the attempt to make this corollary holds that the notion of soft noncommutative schemes via toric geometry is discovered. Let us spell out a guiding question behind the NCS spin-oﬀ of the D-project before leaving this section:

Question 3.12. [soft noncommutative Calabi-Yau space and mirror] What is the correct notion/deﬁnition of soft noncommutative Calabi-Yau spaces in the current context? From pure mathematical generalization of the commutative case? From world-volume conformal in- variance or supersymmetry of D-branes (cf. Sec. 4.2)? What is the mirror symmetry phenomenon in this context?

21 4 (Dynamical) D-branes on a soft noncommutative space

The notion of morphisms from an Azumaya scheme with a fundamental module over C to a soft noncommutative scheme over C is developed in this section.

4.1 Review of Azumaya/matrix schemes with a fundamental module over C The notion of Azumaya schemes (or equivalently matrix schemes) with a fundamental module and morphisms therefrom were developed in [L-Y1] (D(1)) and [L-L-S-Y] (D(2)) in the realm of Algebraic Geometry and in [L-Y2] (D(11.1)) and [L-Y3] (D(11.2)) in the realm of Differential Geometry to capture the Higgsing/unHiggsing feature of D-branes when they coincide or separate and to address dynamical D-branes moving in a space-time, [L-Y4] (D(13.1))and [L-Y5] (D(13.3)). The generalization of such notion to cover fermionic D-branes in Superstring Theory has been a major focus in more recent years; cf. the SUSY spin-off of the D-project, e.g. [L-Y6] (SUSY(2.1)). The key setup of Azumaya/matrix schemes in the realm of Algebraic Geometry that is relevant to the current work is reviewed below for the terminology and notation. Readers are referred to the above works for more details and to [Liu] for a review of the first four years (fall 2007 – fall 2011) of the D-project.

Deﬁnition 4.1.1. [D-brane world volume: Azumaya/matrix scheme over C] Let X be a (Noetherian) scheme over C and E be a locally free OX -module of rank r. Then the sheaf End OX (E) := Hom OX (E, E) of endomorphisms of E is a noncommutative OX -algebra with center OX . The ﬁber of End OX (E) over each C-point on X is isomorphic to the Azumaya algebra (or interchangeably matrix algebra) of r × r matrices over C. E is in the fundamental representation of End OX (E). The new ringed space from the enhanced structure sheaf, with its fundamental representation encoded,

Az Az X := (X, OX := End OX (E), E) . is called an Azumaya scheme (or interchangeably matrix scheme) with a fundamental module. Az The ringed space (X, OX ) serves as the world-volume of a D-brane and the fundamental (left) Az OX -module E serves as the Chan-Paton sheaf on the world-volume. For a complete description of the data on the world-volume of a D-brane, there is also a connection ∇ on E and, in this case, one may introduce a Hermitian structure on E and require that ∇ be unitary. These two additional structures on XAz are irrelevant to the discussion of the current work and hence will be dropped.

Az As the category of (left) OX -modules is equivalent to the category of OX -modules (cf. Morita equivalence), the noncommutativity of XAz may look only of a mild kind — yet important for the correct mathematical description of dynamical D-branes in string theory.

Example 4.1.2. [Azumaya/matrix point with fundamental module/C] To get a feeling of such a mildly noncommutative space, consider the simplest case: X = a C-point p and

Az ⊕r p := (p, Mr(C), C ) , where Mr(C) is the algebra of r × r matrices over C. One may attempt to assign some sen- sible topology “Space (Mr(C))” to the noncommutative C-algebra, in the role of Spec (R) to a commutative ring R. However, all the existing methods only lead to that “Space (Mr(C))”

22 = {p}. Thus, underlying topology is the wrong direction to understand the geometry behind the Azumaya point (interchangeably, matrix point) pAz. Now, the D-brane world-volume is only half of the story. The other half is how the world- volume gets mapped into a target space-time Y (i.e. morphisms from Azumaya schemes to Y ). With this in mind, if A ⊂ Mr(C) is a C-subalgebra and we do know how to make sense of Space (A) and morphisms Space (A) → Y — for example, A is commutative C-subalgebra of Mr(C) and Y is an ordinary Noetherian commutative scheme in Commutative Algebraic Geometry ([Ha]) — then we should expect a morphism pAz → Y from the composition

“Space (Mr(C))” −→−→ Space (A) −→ Y.

Here, −→−→ indicates a dominant morphism since A,→ Mr(C). Az It follows that the correct way to unravel the geometry behind p is through C-subalgebras of Mr(C). For example, for r ≥ 2, Mr(R) contains C-subalgebra in product form A = A1×A2. This Az means p has hidden disconnectivity. Indeed, the consideration of commutative C-subalgebras Az of Mr(C) alone is enough to indicate that p has a very rich geometry behind. Cf. [Liu: Figure 1-2, Figure 3-1, and Figure 3-2].

Example 4.1.2 motivates the following deﬁnition:

Az Az Deﬁnition 4.1.3. [surrogate of X ] (Continuing Deﬁnition 4.1.1.) Let A ⊂ OX be an OX -subalgebra. Then, the ringed space

XA := (X, A)

Az is called a surrogate of X . When A is commutative, XA is simply Spec (A)/X. The inclusion Az OX ⊂ A speciﬁes a dominant morphism XA → X over X while the inclusion A ⊂ OX speciﬁes Az 3 a dominant morphism X → XA over X. Cf. Figure 4-1-1.

4.2 Morphisms from an Azumaya scheme with a fundamental module to a soft noncommutative scheme Y˘ : (Dynamical) D-branes on Y˘ We study in the subsection the notion of morphisms from an Azumaya scheme with a fundamental module to a soft noncommutative scheme Y˘ . The design is guided by the require- ment to reproduce the Higgsing/unHiggsing phenomenon when D-branes collide/coincide or fall apart/separate in a space-time.

Idempotents and gluing of quasi-homomorphisms

Deﬁnition 4.2.1. [quasi-homomorphism of C-algebras] Given two (generally noncommutative) C-algebras R and A, a quasi-homomorphism from R to A is a C-vector-space homomorphism f : R → A such that f(r1r2) = f(r1)f(r2). Note that it is not required that f(1R) = 1A, where 1R and 1A are the identity of R and A respectively. Since f(1R)f(1R) = f(1R), f(1R) is an idempotent of A. Since [f(1R), f(r)] = 0 for all r ∈ R, f(R) ⊂ Centralizer (f(1R)). A C-algebra quasi-homomorphism f : R → S with f(1R) = 1S is called as usual a C-algebra homomorphism. In the other extreme, the zero-map R → 0 ∈ A is a quasi-homomorphism.

3Reproduced from: Chien-Hao Liu and Shing-Tung Yau, D-branes and synthetic/C∞-algebraic symplectic/ calibrated geometry, I. Lemma on a ﬁnite algebraicness property of smooth maps from Azumaya/matrix manifolds, arXiv:1504.01841 [math.SG] (D(12.1)), Figure 1-2.

23 XAz

(a) (c) (b)

X A (g) (d) ( f )

(e)

Figure 4-1-1. The Azumaya/matrix scheme XAz is indicated by a noncommutative cloud sitting over X. In-between are illustrated examples of surrogates (a), (b), (c), Az Az (d), (e), (f), (g) of X . The vertical arrows X → XA and XA → X are the built-in Az dominant morphisms associated to the inclusions OX ⊂ A ⊂ OX of OX -algebras.

Let A, R, S be C-algebras with a fixed C-algebra homomorphism h : R → S. It is standard that two C-algebra homomorphisms fR : R → A and fS : S → A are defined to be glued under h if fR = fS ◦ h. But if, instead, fR and fS are only C-algebra quasi-homomorphisms, it takes some thought as to what the “correct” definition of ‘fR and fS glue under h’ should be. As a generally noncommutative C-algebra, it can happen that A is not a product of C-algebras and yet still contains idempotents other than 0 and 1. In terms of the objects in the opposite category of the category of C-algebras, the geometry Scheme (A) behind the C-algebra A could have hidden disconnectivity (cf. Definition 2.1.8 and Example 2.1.9) and in terms of geometry it can happen that some connected components in the hidden disconnectivity of Scheme (A) is mapped to the complement of the image of Scheme (S) in Scheme (R) under h. Since such hidden disconnectivity of Scheme (A) is inherited from idempotents of A other than 0 and 1, we need to keep track of fR(1R) and fS(1S), allowing the situation fR(1R) 6= fS(1S), as well when considering gluing of C-algebra quasi-homomorphisms fR and fS under h.

Deﬁnition 4.2.2. [subordination relation of idempotents] Let A be a C-algebra and 2 2 e1, e2 ∈ A are idempotents: e1 = e1, e2 = e2. We say that e1 is subordinate to e2, in notation e1 ≺ e2, or equivalently e2 is superior to e1, in notation e2 e1, if

e1 6= e2 and e1e2 = e2e1 = e1 .

Denote ‘e1 ≺ e2 or e1 = e2’ by e1 4 e2 or equivalently e2 < e1. Note that 0 4 e 4 1 for any idempotent e of A.

0 When e1 ≺ e2, e2 := e2 − e1 ∈ A is also an idempotent subordinate to e2 and it satisﬁes 0 0 0 e2e1 = e1e2 = 0. Thus, e2 = e1 + e2 gives an orthogonal decomposition of e2 into subordinate idempotents.

24 Deﬁnition 4.2.3. [gluing of quasi-homomorphisms] Let A, R, S be C-algebras with a ﬁxed C-algebra homomorphism h : R → S and fR : R → A and fS : S → A be quasi-homomorphisms h of C-algebras. We say that fR and fS glue under h, denoted fR fS, if

fS(1S) 4 fR(1R) , fR(R) ⊂ Centralizer (fS(1S)) , and fS(1S) · fR = fR · fS(1S) = fS ◦ h . In terms of a commutative diagram:

fR fS(1S) 4 fR(1R) , fR(R) ⊂ Centralizer (fS(1S)) , and R / A

h fS (1S ) , fS (1S ) · ••·

fS S / A.

Explanation 4.2.4. [gluing of quasi-homomorphisms - geometry behind] Reﬁne the commuting diagram in Deﬁnition 4.2.3 to

fR R / fR(R) / A

h •· fS (1S ) = fS (1S ) ·• =: g fS S / fS(S) / A.

Then, in addition to h, fS and fS are now homomorphisms of C-algebras and so is g, since fS(1S) 4 fR(1R) in A, while the two horizontal inclusions are only quasi-homomorphisms of C-algebras. Let e1 := fR(1R) and e2 := fS(1S) . 0 Then e1 = e1 + e2 is an orthogonal decomposition of e1 by idempotents. Since fR(R) ⊂ Centralizer (e2), 0 fR(R) = fR(R) · e1 + fR(R) · e2 in A 0 is an orthogonal decomposition of the C-algebra fR(R); i.e. fR(R) ' fR(R) · e1 × fR(R) · e2 as abstract C-algebras. Thus, when two quasi-homomorphisms fR : R → A and fS : S → A glue under h in the sense of Deﬁnition 4.2.3, one has a commuting diagram

fR 0 R / fR(R) · e1 × fR(R) · e2 / A

h •· e2 = e2 ·• =: g fS S / fS(S) / A.

Or, equivalently in terms of the objects in the opposite category of the category of C-algebras, a diagram of morphisms of (generally noncommutative) schemes:

0 Scheme (R) o Scheme (fR(R) · e1) q Scheme (fR(R) · e2) O O

Scheme (S) o Scheme (fS(S)) .

Assume in addition that the idempotent e3 := 1 − e1 is also orthogonal to e2 (i.e. e2 e3 = e3 e2 = 0 in A) and let

Ae := {a ∈ A | ae = ea = a}

25 for an idempotent e ∈ A. Note that Ae is a C-algebra with a built-in C-algebra quasi- 0 homomorphism Ae ,→ A. Then, 1 = e1 + e2 + e3 is an orthogonal decomposition of 1 by idempotents in A and

∼ A 0 × A × A −→ A 0 + A + A ,→ A e1 e2 e3 e1 e2 e3 is now an honest -subalgebra inclusion. Since f (R) ⊂ A 0 + A and f (S) ⊂ A , one has C R e1 e2 S e3 now a commuting diagram of C-algebra homomorphisms A 0 × A × A / A e1 e2 e3

π12 fR R / A 0 × A e1 e2

h π2 fS S / Ae2 .

Here π12 and π2 are the projection homomorphisms of product C-algebras to the indicated components. The corresponding commuting diagram of morphisms of noncommutative aﬃne schemes in the opposite category is then: (q = disjoint union)

Scheme (A 0 ) q Scheme (A ) q Scheme (A ) o o Scheme (A) e1 e2 e3 O ι12 ? Scheme (R) o Scheme (A 0 ) q Scheme (A ) e1 e2 O O ι2 ?

Scheme (S) o Scheme (Ae2 ) .

Here, ι12 and ι2 are the inclusion morphism of the connected components as indicated. Thus, a quasi-homomorphism as defined in Definition 4.2.1 is nothing but a roof of ordinary homomorphisms and, with an additional assumption of the completeness of idempotents involved, a gluing of quasi-homomorphisms in the sense of Definition 4.2.3 is nothing but the ordinary gluing of homomorphisms after the replacement by roofs.

Remark 4.2.5. [transitivity of subordination relation of idempotents] Caution that for a general noncommutative C-algebra A, idempotents e1 ≺ e2 and e2 ≺ e3 do not imply e1 ≺ e3. (E.g. 2 2 2 A = Chz1, z2, z2i/(z1 − z1, z2 − z2, z3 − z3, z1z2 − z1, z2z1 − z1, z2z3 − z2, z3z2 − z2).) However, transitivity of ≺ holds for idempotents of the Azumaya/matrix algebra Mr(C) of rank r over C, for all r. (For idempotents e1 ≺ e2, e2 ≺ e3 in Mr(C), the three e1, e2, e3 must be commuting and hence simultaneously diagonizable. Which implies e1 ≺ e3 in the end.)

Homomorphisms from a ∆-system of C-algebras to a C-algebra A

Let ∆ be a fan in NR that satisﬁes Assumption 2.2.2 and A a C-algebra.

A Deﬁnition 4.2.6.[ ∆-system of idempotents in A] Let e∆ := {eσ}σ∈∆ be a collection of A idempotents in A labelled by cones in ∆. e∆ is said to be

26 (i)a weak ∆-system of idempotents in A if eτ 4 eσ for all τ ≺ σ ∈ ∆; 0 (ii)a strong ∆-system of idempotents in A if eσeσ0 = eσ∩σ0 for all σ, σ ∈ ∆. Note that since σ ∩ σ0 = σ0 ∩ σ, all the idempotents in a strong ∆-system of idempotents in A commute with each other. Furthermore, since τ ∩ σ = τ for τ ≺ σ ∈ ∆, a strong ∆-system of idempotents in A is naturally a weak ∆-system of idempotents in A.

A Lemma/Deﬁnition 4.2.7. [reduced idempotents in strong ∆-system] Let e∆ = {eσ}σ∈∆ be a strong ∆-system of idempotents in A. Then, there exists a unique collection {eσ}σ∈∆ of 0 orthogonal (i.e. eσ eσ0 = 0 for σ 6= σ ) idempotents in A such that X eσ = eτ . τ4σ

A eσ is called the reduced idempotent associated to σ ∈ ∆ from the strong ∆-system e∆. It is the contribution to eσ after being trimmed away all the boundary contributions.

Proof. Explicitly, the reduced idempotent eσ, for σ ∈ ∆(k), is given by: X X X e σ := eσ − eτ + eτ − eτ τ∈∆(k−1), τ≺σ τ∈∆(k−2), τ≺σ τ∈∆(k−3), τ≺σ

k−1 X k + ··· + (−1) eτ + (−1) e0 . τ∈∆(1), τ≺σ

Equivalently, let {τ1, ··· , τk} be the set of facets (i.e. codimension-1 faces) of σ ∈ ∆(k), then X X X e = e − e + e e − e e e + ··· + (−1)ke ··· e . σ σ τi τi1 τi2 τi1 τi2 τi3 τ1 τk i i1

A Deﬁnition 4.2.8. [complete strong ∆-system of idempotents] Let e∆ := {eσ}σ∈∆ be a A strong ∆-system of idempotents in A and e∆ := {eσ}σ∈∆ be the associated collection of reduced A P idempotents. We say that e∆ is complete if σ∈∆ eσ = 1.

With all these preparations, we are now ready for a key deﬁnition:

Deﬁnition 4.2.9. [homomorphisms from inverse ∆-system of C-algebras to A] Let ] Q := ({Qσ}σ∈∆, {ιτσ : Qσ → Qτ }τ≺σ∈∆) be an inverse ∆-system of C-algebras. Denote the identity element of Qσ by 1Qσ and that of A by 1. Then a collection

] ] ϕ := {fσ : Qσ → A}σ∈∆ of C-algebra quasi-homomorphisms indexed by cones in ∆ is called a homomorphism from Q to A, denoted now ϕ] : Q → A, if

27 (i)( Inverse ∆-system of quasi-homomorphisms of C-algebras)

] ιτσ fσ fτ , for τ ≺ σ ∈ ∆ , ] Cf. Deﬁnition 4.2.3. In particular, eϕ] := {eσ := fσ(1Qσ )}σ∈∆ is a weak ∆-system of idempotents in A, called the ∆-system of idempotents in A associated to ϕ].

(ii)( Completeness) eϕ] is a complete strong ∆-system of idempotents in A. For the simple uniformness of terminology but with slight abuse, we shall call such homomorphisms also C-algebra homomorphisms when need to distinguish with homomorphisms of other algebraic structures. The set of homomorphisms from Q to A is thus denoted Hom C-Alg (Q,A), or simply Hom (Q,A).

Explanation 4.2.10. [gluings of ∆-system of (local) quasi-homomorphisms to a “global” homomorphism] Given a homomorphism ϕ] : Q → A as deﬁned in Deﬁni- ] tion 4.2.9. Let eϕ] := {eσ}σ∈∆, eσ := fσ(1Qσ ), σ ∈ ∆, be the collection of the reduced idempotents from eϕ] and

Aeσ := {a ∈ A | aeeσ = eeσ a = a} , σ ∈ ∆ . Then, a generalization of Explanation 4.2.4 gives the inclusions of C-algebras f ] (Q ) ⊂ ⊕ A (as a multiplicatively closed -vector subspace of A) σ σ τ4σ eτ C ' × A (as an abstract -algebra) , σ ∈ ∆ , τ4σ eτ C and commuting diagrams of homomorphisms of C-algebras:

π] Aϕ Aϕ := ×σ∈∆Aeσ / A

] σ f ] Q σ A := × A σ / ϕ, σ ρ4σ∈∆ eρ

] ] ιτσ τσ f ] Q τ A := × A , for τ ≺ σ ∈ ∆ . τ / ϕ, τ ρ4τ∈∆ eρ

] ] Here σ : Aϕ → Aϕ,σ, σ ∈ ∆, and τσ : Aϕ, σ → Aϕ, τ are the projection maps onto factors of product C-algebras as indicated. The underlying geometry of the diagrams is revealed in terms of commuting diagrams of morphisms of noncommutative aﬃne schemes in the opposite category: (q = disjoint union)

πAϕ Scheme (Aϕ) = qσ∈∆Scheme (Aeσ ) o o Scheme (A) O σ ? f Scheme (Q ) σ Scheme (A ) = q Scheme (A ) σ o ϕ, σ ρ4σ∈∆ eρ O O ιτσ τσ ? f Scheme (Q ) τ Scheme (A ) = q Scheme (A ) , τ o ϕ, τ ρ4τ∈∆ eρ for τ ≺ σ ∈ ∆. Think of Q as deﬁning contravariantly a noncommutative space “Space (Q)”. ϕ] then deﬁnes a morphism ϕ : Scheme (A) → Space (Q) of noncommutative spaces.

28 (Dynamical, complex algebraic) D-branes on a soft noncommutative scheme

Let X be a (commutative) scheme over C, E be a locally free OX -module of rank r, Az Az X = (X, OX := End OX (E), E) be the Azumaya/matrix scheme over C with the underlying topology X and the fundamental module E, and Y˘ be a soft noncommutative scheme via toric geometry, with the structure sheaf ˘ ˘ ] OY˘ = R∆ := ({Rσ}σ∈∆, {ιτσ}τ≺σ∈∆) . an inverse ∆-system of C-algebras, cf. Deﬁnition 3.10. Consider the sheaf of sets Az Hom C-Alg (OY˘ , OX ) over X ˘ Az associated to the presheaf deﬁned by the assignment U 7→ Hom C-Alg (R∆, OX (U)) for U open ˘ Az ˘ Az in X and restriction maps Hom C-Alg (R∆, OX (U)) → Hom C-Alg (R∆, OX (V )) via the post- Az Az composition with the C-algebra homomorphism OX (U) → OX (V ) for open sets V ⊂ U ⊂ X.

Deﬁnition 4.2.11. [D-brane on Y˘ : morphism XAz → Y˘ ] A morphism ϕ : XAz → Y˘ is by deﬁnition a global section, in notation

] Az ϕ : OY˘ −→ OX Az ] of Hom C-Alg (OY˘ , OX ). Explicitly, ϕ is represented by the following data ] ] ˘ (U := {Uα}α∈I , ϕU := {ϕα : R∆ → End OX (E|Uα )}α∈I ) , ] where U is an open covering of X and ϕU a collection of C-algebra homomorphisms such that the following diagrams commute

End O (E|Uα ) h X k H ] ϕα

v ˘ End OX (E|Uα∩Uβ ) R∆ h

ϕ] 6 V s β End OX (E|Uβ ) , for all α, β ∈ I. With the additional data of a connection ∇ on E (the Chan-Paton sheaf with a gauge ﬁeld) this gives a mathematical model for dynamical, complex algebraic D-branes as an extended object moving on the soft noncommutative target-space Y˘ over C. (One may also introduce a Hermitian structure on E and consider only unitary connections ∇ on E.)

Deﬁnition 4.2.12. [surrogate of XAz associated to ϕ] The completeness condition on idempotents (cf. Deﬁnition 4.2.9: Condition (ii)) implies that the inclusion ] Az Aϕ := OX hϕ (OY˘ )i ,→ OX is an OX -algebra homomorphism (rather than only a quasi- homomorphism of OX -algebras). The corresponding noncommutative scheme Xϕ over X is π] Az ϕ called the surrogate of X associated to ϕ. The sequence of OX -algebra inclusions OX ,→ Az Az πϕ Aϕ ,→ OX gives a sequence of dominant morphisms X −→−→ Xϕ −→−→ X. By construction, ϕ Az fϕ factors to a composition of X →→ Xϕ −→ Y˘ .

29 Deﬁnition 4.2.13. [image of ϕ and push-forward Chan-Paton sheaf] The two-sided ] ˘ ideal sheaf Ker (ϕ ) of OY˘ deﬁnes a soft closed subscheme of Y , denoted Im ϕ and called the ] image of ϕ. ϕ renders E an OY˘ -module, denoted ϕ∗E and called the push-forward Chan-Paton sheaf on Y˘ . By construction, ϕ∗E is supported on Im ϕ.

Figure 4-2-1.4

XAz

ϕ Y

Xϕ fϕ

�ϕ Imϕ

Figure 4-2-1. A morphism ϕ : XAz → Y˘ is illustrated. Note that the image ϕ(XAz) of ϕ in Y can have very rich variations of behavior as a closed subscheme of Y˘ .

We conclude this subsection with a second guiding question:

r Question 4.2.14. [generalized matrix model] For X a C-point pt, E = C , and ∆ = {σ}, ˘ n where σ is a simplicial cone of NR of index 1, one has that Y (∆) = ncA and morphisms Az n from pt to ncA are given by tuples m = (m1, ··· , mn), mi ∈ Mr(C). Such m’s (with possible Hermitian constraints) are ﬁelds in a matrix model. From this aspect, a theory with ﬁelds morphisms from ptAz to a soft noncommutative toric scheme Y˘ (∆) for a fan ∆ satisfying Assumption 2.2.2 gives a generalization of matrix models via gluing simple matrix models on morphisms ptAz → ncAn. Details of some examples? Consequences? Relations to Topological String Theory?

4.3 Two equivalent descriptions of a morphism ϕ : XAz → Y˘

The notion of a soft noncommutative toric scheme Y˘ (∆) over C associated to a fan ∆ can be generalized to the notion of a soft noncommutative toric scheme Y˘X (∆) over X associated ˘ to ∆ by generalizing the notion of an inverse ∆-system of C monoid algebras {ChMσi}σ∈∆ to the notion of an inverse ∆-system of OX monoid algebras {OX hM˘ σi}σ∈∆. Similarly, for close

4Revised from: Chien-Hao Liu and Shing-Tung Yau, More on the admissible condition on diﬀerentiable Az maps ϕ :(X ,E; ∇) → Y in the construction of the non-Abelian Dirac-Born-Infeld action SDBI (ϕ, ∇), arXiv:1611.09439 [hep-th] (D(13.2.1)), Figure 2-1.

30 subschemes Z˘ of Y˘ . This deﬁnes the product space X × Y˘ of the commutative scheme X and ˘ the soft noncommutative scheme Y and its structure sheaf OX×Y˘ . The built-in inclusion

OX ,→ Center (OX×Y˘ ) ⊂ OX×Y˘ ˘ of OX -algebras deﬁnes the projection map prX : X ×Y → X. In particular, an OX×Y˘ -module Fe ˘ on X × Y is automatically an OX -module, denoted prX, ∗ Fe or simply Fe. The built-in inclusion

OY˘ ,→ OY˘ · 1 ⊂ OX×Y˘ ˘ ˘ of C-algebras deﬁnes the projection map prY˘ : X × Y → Y . Let Fe be a two-sided OX×Y˘ -module. The support of Fe, in notation Supp (Fe), is by deﬁnition ˘ the closed subscheme of X × Y associated to the two-sided ideal sheaf of OX×Y˘ generated by both left annihilators and right annihilators of Fe

Ann (Fe) := (f ∈ OX×Y˘ | f · Fe = 0 or F·e f = 0) .

We say that Fe is of relative dimension 0 over X if the restriction of Fe to each {p} × Y˘ for p a -point on X is a ﬁnitely dimensional -vector space and O := O /Ann (F) is a C C Supp (Fe) X×Y˘ e coherant OX -module under prX, ∗.

Definition/Lemma 4.3.1. [graph of morphism ϕ : XAz → Y˘ ] Let ϕ : XAz → Y˘ be a ] Az ] morphism, defined via ϕ : OY˘ → OX := End OX (E). Then, ϕ determines a homomorphism of OX -algebras ] Az ] ϕe : OX×Y˘ −→ OX , with f · r˘ 7−→ f · ϕ (˘r) for all f ∈ OX andr ˘ ∈ OY˘ . Az ˘ ] This defines the morphism ϕe : X → X ×Y . The two-sided ideal sheaf Ker (ϕe ) of OX×Y˘ defines Az a closed subscheme Γϕ of X × Y˘ that is isomorphic to the surrogate Xϕ of X associated to ϕ, as noncommutative schemes over X. By construction, Ee := ϕe∗E is an OX×Y˘ -module supported on Γϕ. The OX×Y˘ -module Ee is called the graph of ϕ. It has the property that

· Ee is of relative dimension 0 over X and prX, ∗Ee = E (i.e. prX, ∗E'Ee canonically).

The converse also holds: Let Ee be an OX×Y˘ -module of relative dimension 0 over X such that E := prX, ∗Ee is a coherent locally free OX -module, say of rank r and Az Az X := (X, OX := End OX (E), E) be the Azumaya scheme/C with the fundamental module E. Then, Ee speciﬁes a morphism ϕ : XAz → Y˘ . Cf. [L-L-S-Y: Figure 2-2-1] (D(2)).

Proof. Via pr] , the O -module structure on E induces a O -module structure on E. This Y˘ Y˘ e Y˘ ] Az deﬁnes a homomorphism ϕ : OY˘ → OX .

From Deﬁnition/Lemma 4.3.1, one realizes that

0;r ˘ Lemma 4.3.2.[ ϕ and moduli stack of 0-dimensional OY˘ -modules] Let M (Y ) be the moduli stack of 0-dimensional OY˘ -modules that are of complex dimension r as C-vector spaces. Then a morphism from an Azumaya scheme over X with the fundamental module of rank r to Y˘ gives a morphism X → M0;r(Y˘ ). Cf. [L-L-S-Y: Figure 3-1-1] (D(2)).

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